answer：First, I need to determine the total length of the race, which I'll denote as ( L ) meters.When Harper finished the race, Jack was 152 meters from the start line. This means that Harper was 152 meters ahead of Jack at that point.Since Harper and Jack were 848 meters apart when Harper finished, I can set up the equation:[L - 152 = 848]To find ( L ), I'll solve the equation:[L = 848 + 152][L = 1000]Therefore, the total length of the race is 1000 meters.

answer：Okay, so I need to find the minimum value of the expression (9x + frac{1}{x^6}) where (x > 0). Hmm, I remember that for optimization problems like this, calculus is usually helpful. Maybe I can use derivatives to find the minimum. Let me think through this step by step.First, let me write down the function I need to minimize:[ f(x) = 9x + frac{1}{x^6} ]Since (x > 0), I don't have to worry about the function being undefined or anything like that.To find the minimum, I should find the critical points of this function. Critical points occur where the derivative is zero or undefined. So, I need to compute the derivative of (f(x)) with respect to (x).Let's compute (f'(x)):[ f'(x) = frac{d}{dx} left(9x + frac{1}{x^6}right) ]The derivative of (9x) is straightforward—it's just 9. For the second term, (frac{1}{x^6}), I can rewrite it as (x^{-6}) to make differentiation easier. The derivative of (x^{-6}) is (-6x^{-7}), which is (-frac{6}{x^7}).So putting it all together:[ f'(x) = 9 - frac{6}{x^7} ]Now, I need to find where this derivative equals zero because that's where the function could have a minimum or maximum.Set (f'(x) = 0):[ 9 - frac{6}{x^7} = 0 ]Let me solve for (x). First, move the fraction to the other side:[ 9 = frac{6}{x^7} ]Now, multiply both sides by (x^7) to eliminate the denominator:[ 9x^7 = 6 ]Divide both sides by 9:[ x^7 = frac{6}{9} ]Simplify the fraction:[ x^7 = frac{2}{3} ]To solve for (x), take the seventh root of both sides:[ x = left(frac{2}{3}right)^{frac{1}{7}} ]Hmm, that's a bit messy, but it's a valid expression. Let me see if I can write it in a nicer form or approximate it, but maybe I don't need to compute it numerically right now.Okay, so I have a critical point at (x = left(frac{2}{3}right)^{frac{1}{7}}). Now, I need to check whether this critical point is a minimum or a maximum. Since the function is defined for (x > 0), and as (x) approaches 0, (frac{1}{x^6}) goes to infinity, and as (x) approaches infinity, (9x) also goes to infinity. So, the function tends to infinity on both ends, which suggests that this critical point is likely a minimum.But just to be thorough, I can use the second derivative test to confirm whether this critical point is a minimum.Let's compute the second derivative (f''(x)). Starting from the first derivative:[ f'(x) = 9 - frac{6}{x^7} ]Differentiate again:[ f''(x) = 0 - frac{d}{dx}left(6x^{-7}right) ]The derivative of (6x^{-7}) is (-42x^{-8}), so:[ f''(x) = 42x^{-8} = frac{42}{x^8} ]Since (x > 0), (x^8) is always positive, so (f''(x)) is positive. A positive second derivative at a critical point means that the function is concave up there, which indicates a local minimum. Therefore, the critical point we found is indeed a minimum.Now, I need to find the value of the function at this critical point to get the minimum value.So, substitute (x = left(frac{2}{3}right)^{frac{1}{7}}) into (f(x)):[ fleft(left(frac{2}{3}right)^{frac{1}{7}}right) = 9left(frac{2}{3}right)^{frac{1}{7}} + frac{1}{left(left(frac{2}{3}right)^{frac{1}{7}}right)^6} ]Let me simplify this expression step by step.First, let's compute the first term:[ 9left(frac{2}{3}right)^{frac{1}{7}} ]I can leave it as is for now.Now, the second term:[ frac{1}{left(left(frac{2}{3}right)^{frac{1}{7}}right)^6} ]Simplify the exponent:[ left(frac{2}{3}right)^{frac{6}{7}} ]So, the second term becomes:[ left(frac{3}{2}right)^{frac{6}{7}} ]Therefore, the function value at the critical point is:[ 9left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} ]Hmm, this looks a bit complicated. Maybe there's a way to express this more neatly or perhaps recognize a pattern. Alternatively, I could try to express both terms with the same exponent to see if they can be combined.Let me note that (left(frac{2}{3}right)^{frac{1}{7}}) is the same as (left(frac{2}{3}right)^{frac{1}{7}}), and (left(frac{3}{2}right)^{frac{6}{7}}) is the reciprocal raised to the 6th power. Maybe I can write both terms with the same base.Let me denote (y = left(frac{2}{3}right)^{frac{1}{7}}). Then, the first term is (9y), and the second term is (left(frac{3}{2}right)^{frac{6}{7}} = left(left(frac{2}{3}right)^{-1}right)^{frac{6}{7}} = left(frac{2}{3}right)^{-frac{6}{7}} = y^{-6}).So, the function value becomes:[ 9y + y^{-6} ]Hmm, that's interesting. So, (f(x)) at the critical point is (9y + y^{-6}), where (y = left(frac{2}{3}right)^{frac{1}{7}}). I wonder if there's a way to express this in terms of a single exponent or if it simplifies further.Alternatively, maybe I can write both terms with exponent (frac{1}{7}) to see if they can be combined.Let me try that. Since (y = left(frac{2}{3}right)^{frac{1}{7}}), then (y^7 = frac{2}{3}).So, (y^{-6} = left(y^7right)^{-frac{6}{7}} = left(frac{2}{3}right)^{-frac{6}{7}} = left(frac{3}{2}right)^{frac{6}{7}}), which is what we had before.Hmm, not sure if that helps. Maybe I can express (9y) in terms of (y^7). Since (y^7 = frac{2}{3}), then (y = left(frac{2}{3}right)^{frac{1}{7}}), so (9y = 9left(frac{2}{3}right)^{frac{1}{7}}).Alternatively, perhaps I can factor out (y^{-6}) or something like that. Let me see:[ 9y + y^{-6} = y^{-6} + 9y ]Not sure if that helps. Maybe I can write it as:[ y^{-6} + 9y = y^{-6} + 9y ]Hmm, not particularly helpful.Alternatively, maybe I can write both terms with exponent (frac{1}{7}). Let me see:Since (y = left(frac{2}{3}right)^{frac{1}{7}}), then (y^k = left(frac{2}{3}right)^{frac{k}{7}}).So, (9y = 9left(frac{2}{3}right)^{frac{1}{7}}) and (y^{-6} = left(frac{2}{3}right)^{-frac{6}{7}} = left(frac{3}{2}right)^{frac{6}{7}}).Alternatively, perhaps I can write both terms with exponent (frac{1}{7}):[ 9left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} = 9left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} ]Hmm, maybe I can factor out (left(frac{2}{3}right)^{frac{1}{7}}) from both terms? Let's see:[ 9left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} = left(frac{2}{3}right)^{frac{1}{7}} left(9 + left(frac{3}{2}right)^{frac{7}{7}}right) ]Wait, because (left(frac{3}{2}right)^{frac{6}{7}} = left(frac{3}{2}right)^{frac{7}{7} - frac{1}{7}} = left(frac{3}{2}right)^{1 - frac{1}{7}} = left(frac{3}{2}right) cdot left(frac{3}{2}right)^{-frac{1}{7}}).But I'm not sure if that helps. Alternatively, maybe I can write (left(frac{3}{2}right)^{frac{6}{7}} = left(frac{3}{2}right)^{frac{6}{7}}), which is the same as (left(frac{3}{2}right)^{frac{6}{7}}).Alternatively, perhaps I can write both terms with the same exponent. Let me see:Let me denote (k = frac{1}{7}), so (y = left(frac{2}{3}right)^k). Then, (y^{-6} = left(frac{2}{3}right)^{-6k} = left(frac{3}{2}right)^{6k}).So, the function value is:[ 9left(frac{2}{3}right)^k + left(frac{3}{2}right)^{6k} ]Where (k = frac{1}{7}).Hmm, not sure if that helps either. Maybe I can think of this as a function in terms of (k), but that might be overcomplicating things.Alternatively, perhaps I can use logarithms to simplify the expression. Let me take the natural logarithm of both terms and see if that helps, but I'm not sure if that's necessary.Wait, maybe I can express both terms in terms of (left(frac{2}{3}right)^{frac{1}{7}}). Let me see:We have:[ 9left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} ]Notice that (left(frac{3}{2}right)^{frac{6}{7}} = left(frac{2}{3}right)^{-frac{6}{7}} = left(frac{2}{3}right)^{frac{1}{7} times (-6)} = left(left(frac{2}{3}right)^{frac{1}{7}}right)^{-6}).So, if I let (y = left(frac{2}{3}right)^{frac{1}{7}}), then the function value is:[ 9y + y^{-6} ]Hmm, that's the same as before. Maybe I can write this as:[ 9y + frac{1}{y^6} ]Which is the original function (f(y)). Wait, that's interesting. So, substituting (x = y) into the original function gives me the same expression. That seems a bit circular.Alternatively, maybe I can consider that at the critical point, the function value is equal to (9y + y^{-6}), and perhaps there's a relationship between these terms that can be exploited.Wait, let's think about the AM-GM inequality. The Arithmetic Mean-Geometric Mean inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean, with equality when all the numbers are equal.In this case, I have two terms: (9x) and (frac{1}{x^6}). But AM-GM typically applies to sums of terms, so maybe I can split the (9x) into nine terms of (x) each, and the (frac{1}{x^6}) as one term. That way, I have a total of 10 terms.So, let me try that. Let me write:[ 9x + frac{1}{x^6} = x + x + x + x + x + x + x + x + x + frac{1}{x^6} ]Now, I have 10 terms: nine (x)s and one (frac{1}{x^6}).Applying AM-GM to these 10 terms:[ frac{x + x + x + x + x + x + x + x + x + frac{1}{x^6}}{10} geq sqrt[10]{x cdot x cdot x cdot x cdot x cdot x cdot x cdot x cdot x cdot frac{1}{x^6}} ]Simplify the right-hand side:[ sqrt[10]{x^9 cdot frac{1}{x^6}} = sqrt[10]{x^{3}} = x^{frac{3}{10}} ]So, the inequality becomes:[ frac{9x + frac{1}{x^6}}{10} geq x^{frac{3}{10}} ]Multiply both sides by 10:[ 9x + frac{1}{x^6} geq 10x^{frac{3}{10}} ]Hmm, but I'm not sure if this helps me find the minimum value directly. It gives me a lower bound in terms of (x), but I need a numerical value.Wait, maybe I can find the value of (x) that makes the equality hold in AM-GM. Equality holds when all the terms are equal. So, in this case, all ten terms must be equal:[ x = x = x = x = x = x = x = x = x = frac{1}{x^6} ]So, (x = frac{1}{x^6}). Let's solve for (x):[ x = frac{1}{x^6} ]Multiply both sides by (x^6):[ x^7 = 1 ]So, (x = 1), since (x > 0).Wait, but earlier, using calculus, I found that the critical point is at (x = left(frac{2}{3}right)^{frac{1}{7}}), which is approximately (0.905), not 1. So, there's a discrepancy here. That suggests that either my AM-GM approach is incorrect or I made a mistake in the calculus.Let me double-check my calculus. I had:[ f'(x) = 9 - frac{6}{x^7} ]Setting this equal to zero:[ 9 = frac{6}{x^7} ][ x^7 = frac{6}{9} = frac{2}{3} ][ x = left(frac{2}{3}right)^{frac{1}{7}} ]That seems correct. So, why does the AM-GM approach suggest that the minimum occurs at (x = 1)?Wait, maybe I misapplied the AM-GM inequality. Let me think again. When I split (9x) into nine (x)s and one (frac{1}{x^6}), I have ten terms. For AM-GM to give the tightest bound, all terms should be equal. So, setting (x = frac{1}{x^6}) gives (x^7 = 1), so (x = 1). But according to calculus, the minimum occurs at (x = left(frac{2}{3}right)^{frac{1}{7}}), which is not 1.This suggests that the AM-GM approach might not be directly applicable here, or perhaps I need to adjust the number of terms or their weights.Alternatively, maybe I can use weighted AM-GM. Since the coefficients of the terms are different, perhaps I need to assign weights accordingly.Let me recall that the weighted AM-GM inequality states that for positive real numbers (a_i) and positive weights (w_i) with (sum w_i = 1), we have:[ sum w_i a_i geq prod a_i^{w_i} ]Equality holds when (a_i) are proportional to (w_i).In our case, we have two terms: (9x) and (frac{1}{x^6}). Let me assign weights to these terms. Let me denote (w_1) as the weight for (9x) and (w_2) as the weight for (frac{1}{x^6}). Since there are two terms, we can set (w_1 + w_2 = 1).But I'm not sure if this is the right approach. Alternatively, maybe I can consider the exponents in the terms. The term (9x) has an exponent of 1, and (frac{1}{x^6}) has an exponent of -6. Maybe I can use the method of Lagrange multipliers or another optimization technique.Wait, perhaps I can use the method of setting the derivative to zero, which I already did, and then express the function value in terms of (x). Let me go back to that.We have:[ x = left(frac{2}{3}right)^{frac{1}{7}} ]So, (x^7 = frac{2}{3}).Now, let's compute (f(x)) at this point:[ f(x) = 9x + frac{1}{x^6} ]Express (x^6) in terms of (x^7):Since (x^7 = frac{2}{3}), then (x^6 = frac{x^7}{x} = frac{2}{3x}).So, (frac{1}{x^6} = frac{3x}{2}).Therefore, substituting back into (f(x)):[ f(x) = 9x + frac{3x}{2} = left(9 + frac{3}{2}right)x = frac{21}{2}x ]But (x = left(frac{2}{3}right)^{frac{1}{7}}), so:[ f(x) = frac{21}{2} left(frac{2}{3}right)^{frac{1}{7}} ]Hmm, that's a bit simpler, but it's still not a nice number. Maybe I can express it in terms of exponents or see if it can be simplified further.Alternatively, perhaps I can write it as:[ f(x) = frac{21}{2} cdot left(frac{2}{3}right)^{frac{1}{7}} ]Which is approximately:[ frac{21}{2} approx 10.5 ]And (left(frac{2}{3}right)^{frac{1}{7}}) is approximately (0.905), so:[ 10.5 times 0.905 approx 9.5025 ]But wait, earlier when I used AM-GM, I got a lower bound of 10, but according to calculus, the minimum is approximately 9.5, which is lower than 10. That suggests that the AM-GM approach gave a lower bound that's not tight, meaning the actual minimum is lower than that bound.So, perhaps the AM-GM approach isn't the best way to go here, and calculus gives a more accurate result.Alternatively, maybe I can use substitution to simplify the expression. Let me set (t = x^7), so (x = t^{frac{1}{7}}). Then, (x^6 = t^{frac{6}{7}}), and (frac{1}{x^6} = t^{-frac{6}{7}}).So, the function becomes:[ f(t) = 9t^{frac{1}{7}} + t^{-frac{6}{7}} ]Now, let me compute the derivative of (f(t)) with respect to (t):[ f'(t) = 9 cdot frac{1}{7} t^{-frac{6}{7}} - frac{6}{7} t^{-frac{13}{7}} ]Set this equal to zero:[ frac{9}{7} t^{-frac{6}{7}} - frac{6}{7} t^{-frac{13}{7}} = 0 ]Multiply both sides by (7 t^{frac{13}{7}}) to eliminate denominators:[ 9 t^{frac{7}{7}} - 6 = 0 ]Simplify:[ 9t - 6 = 0 ][ 9t = 6 ][ t = frac{6}{9} = frac{2}{3} ]So, (t = frac{2}{3}), which means (x = left(frac{2}{3}right)^{frac{1}{7}}), confirming our earlier result.Now, substituting back into (f(t)):[ fleft(frac{2}{3}right) = 9 left(frac{2}{3}right)^{frac{1}{7}} + left(frac{2}{3}right)^{-frac{6}{7}} ]Which is the same as:[ 9 left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} ]As before.Hmm, so it seems that regardless of the substitution, I end up with the same expression for the minimum value. It doesn't seem to simplify nicely into a whole number or a simple fraction. So, perhaps the minimum value is indeed (frac{21}{2} left(frac{2}{3}right)^{frac{1}{7}}), which is approximately 9.5.But wait, earlier when I used AM-GM, I got a lower bound of 10, which is higher than this value. That suggests that the AM-GM approach isn't giving the tightest possible bound in this case, which is fine because AM-GM provides a lower bound, not necessarily the exact minimum.Alternatively, maybe I can express the minimum value in terms of exponents with a common base. Let me see:We have:[ f(x) = 9x + frac{1}{x^6} ]At the critical point (x = left(frac{2}{3}right)^{frac{1}{7}}), so:[ f(x) = 9 left(frac{2}{3}right)^{frac{1}{7}} + left(frac{3}{2}right)^{frac{6}{7}} ]Let me write both terms with the same exponent:[ 9 left(frac{2}{3}right)^{frac{1}{7}} = 9 cdot left(frac{2}{3}right)^{frac{1}{7}} ][ left(frac{3}{2}right)^{frac{6}{7}} = left(frac{2}{3}right)^{-frac{6}{7}} ]So, combining these:[ f(x) = 9 cdot left(frac{2}{3}right)^{frac{1}{7}} + left(frac{2}{3}right)^{-frac{6}{7}} ]Let me factor out (left(frac{2}{3}right)^{-frac{6}{7}}):[ f(x) = left(frac{2}{3}right)^{-frac{6}{7}} left(9 cdot left(frac{2}{3}right)^{frac{1}{7} + frac{6}{7}} + 1right) ]Simplify the exponent:[ frac{1}{7} + frac{6}{7} = 1 ]So:[ f(x) = left(frac{2}{3}right)^{-frac{6}{7}} left(9 cdot left(frac{2}{3}right) + 1right) ]Compute (9 cdot left(frac{2}{3}right)):[ 9 cdot frac{2}{3} = 6 ]So:[ f(x) = left(frac{2}{3}right)^{-frac{6}{7}} (6 + 1) = 7 cdot left(frac{2}{3}right)^{-frac{6}{7}} ]Which is:[ 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Hmm, that's another way to write it, but it's still not a simple number.Alternatively, perhaps I can express this in terms of (x). Since (x^7 = frac{2}{3}), then (left(frac{3}{2}right)^{frac{6}{7}} = left(x^7right)^{-frac{6}{7}} = x^{-6}). So, the function value is:[ 7x^{-6} ]But (x^{-6} = frac{1}{x^6}), which is part of the original function. So, this doesn't seem to help.Alternatively, maybe I can write the function value as:[ 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Which is approximately:[ 7 times 1.147 = 8.029 ]Wait, that can't be right because earlier I had approximately 9.5. Wait, let me check my calculations.Wait, (left(frac{3}{2}right)^{frac{6}{7}}) is approximately:First, compute (lnleft(frac{3}{2}right) approx 0.4055).Then, multiply by (frac{6}{7}):[ 0.4055 times frac{6}{7} approx 0.4055 times 0.8571 approx 0.347 ]Then, exponentiate:[ e^{0.347} approx 1.414 ]So, (left(frac{3}{2}right)^{frac{6}{7}} approx 1.414).Then, (7 times 1.414 approx 9.898), which is approximately 9.9, which is close to my earlier approximation of 9.5. Hmm, perhaps my earlier approximation was a bit off.Wait, let me compute (left(frac{3}{2}right)^{frac{6}{7}}) more accurately.First, compute (lnleft(frac{3}{2}right) approx 0.4054651).Multiply by (frac{6}{7}):[ 0.4054651 times frac{6}{7} approx 0.4054651 times 0.8571429 approx 0.3473 ]Now, compute (e^{0.3473}):Using Taylor series or a calculator approximation:(e^{0.3473} approx 1 + 0.3473 + frac{0.3473^2}{2} + frac{0.3473^3}{6} + frac{0.3473^4}{24})Compute each term:1. (1)2. (0.3473)3. (frac{0.3473^2}{2} = frac{0.1206}{2} = 0.0603)4. (frac{0.3473^3}{6} = frac{0.0419}{6} approx 0.00698)5. (frac{0.3473^4}{24} = frac{0.0145}{24} approx 0.000604)Adding these up:[ 1 + 0.3473 = 1.3473 ][ 1.3473 + 0.0603 = 1.4076 ][ 1.4076 + 0.00698 = 1.4146 ][ 1.4146 + 0.000604 approx 1.4152 ]So, (e^{0.3473} approx 1.4152), which is very close to (sqrt{2} approx 1.4142). So, (left(frac{3}{2}right)^{frac{6}{7}} approx 1.4152).Therefore, (7 times 1.4152 approx 9.9064), which is approximately 9.906.But earlier, when I computed (f(x)) as (frac{21}{2} times left(frac{2}{3}right)^{frac{1}{7}}), I approximated (left(frac{2}{3}right)^{frac{1}{7}}) as 0.905, so:[ frac{21}{2} times 0.905 approx 10.5 times 0.905 approx 9.5025 ]Which is about 9.5, which is a bit lower than 9.906. Hmm, that's inconsistent. I must have made a mistake in one of the approximations.Wait, let me compute (left(frac{2}{3}right)^{frac{1}{7}}) more accurately.Compute (lnleft(frac{2}{3}right) approx -0.4054651).Multiply by (frac{1}{7}):[ -0.4054651 times frac{1}{7} approx -0.0579236 ]Now, compute (e^{-0.0579236}):Using Taylor series:(e^{-0.0579236} approx 1 - 0.0579236 + frac{0.0579236^2}{2} - frac{0.0579236^3}{6})Compute each term:1. (1)2. (-0.0579236)3. (frac{0.003354}{2} = 0.001677)4. (-frac{0.000194}{6} approx -0.0000323)Adding these up:[ 1 - 0.0579236 = 0.9420764 ][ 0.9420764 + 0.001677 = 0.9437534 ][ 0.9437534 - 0.0000323 approx 0.9437211 ]So, (left(frac{2}{3}right)^{frac{1}{7}} approx 0.9437).Therefore, (frac{21}{2} times 0.9437 approx 10.5 times 0.9437 approx 9.899), which is approximately 9.9, matching the earlier calculation.So, both methods give me approximately 9.9, which is consistent. Therefore, the minimum value is approximately 9.9.But the problem asks for the exact minimum value, not an approximation. So, I need to express it in terms of exponents or radicals.From earlier, we have:[ f(x) = 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Alternatively, since (x = left(frac{2}{3}right)^{frac{1}{7}}), we can write:[ f(x) = 7 cdot left(frac{3}{2}right)^{frac{6}{7}} = 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Which is the exact value.Alternatively, we can write it as:[ f(x) = 7 cdot left(frac{3}{2}right)^{frac{6}{7}} = 7 cdot left(frac{3^6}{2^6}right)^{frac{1}{7}} = 7 cdot left(frac{729}{64}right)^{frac{1}{7}} ]But that might not be simpler.Alternatively, perhaps we can write it as:[ f(x) = 7 cdot left(frac{3}{2}right)^{frac{6}{7}} = 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Which is as simplified as it gets.Alternatively, since (x^7 = frac{2}{3}), we can write:[ f(x) = 9x + frac{1}{x^6} = 9x + frac{x}{2} = frac{19x}{2} ]Wait, that doesn't make sense because earlier we had (x^6 = frac{2}{3x}), so (frac{1}{x^6} = frac{3x}{2}), so:[ f(x) = 9x + frac{3x}{2} = frac{21x}{2} ]But (x = left(frac{2}{3}right)^{frac{1}{7}}), so:[ f(x) = frac{21}{2} cdot left(frac{2}{3}right)^{frac{1}{7}} ]Which is another way to write it.So, in conclusion, the exact minimum value is:[ frac{21}{2} cdot left(frac{2}{3}right)^{frac{1}{7}} ]Or equivalently:[ 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Both expressions are correct and represent the exact minimum value.Alternatively, perhaps I can rationalize it further or express it in terms of radicals, but I don't think it simplifies nicely into a whole number or a simple fraction. Therefore, the minimum value is best expressed in terms of exponents as above.So, to summarize, using calculus, I found that the function (f(x) = 9x + frac{1}{x^6}) has a critical point at (x = left(frac{2}{3}right)^{frac{1}{7}}), which is a minimum since the second derivative is positive there. The minimum value is then:[ fleft(left(frac{2}{3}right)^{frac{1}{7}}right) = frac{21}{2} cdot left(frac{2}{3}right)^{frac{1}{7}} ]Or equivalently:[ 7 cdot left(frac{3}{2}right)^{frac{6}{7}} ]Which is approximately 9.9.Therefore, the minimum value of the expression is (boxed{frac{21}{2} left(frac{2}{3}right)^{frac{1}{7}}}).

answer：Okay, so I've got this sequence problem here. Let me read it again to make sure I understand what's being asked. It says, "Given a sequence {a_n} where a_n ≠ 0, a₁ = 1, and 1/a_{n+1} = 1/a_n + 2, find the value of a_{20}." Alright, so I need to figure out what the 20th term of this sequence is.First, let me parse the given information. The sequence is defined such that each term a_n is not zero, which is good because we're dealing with reciprocals here. The first term is 1, so a₁ is 1. Then, there's this recurrence relation: 1/a_{n+1} equals 1/a_n plus 2. So, each subsequent reciprocal term is the previous reciprocal plus 2.Hmm, that sounds like an arithmetic sequence. Because in an arithmetic sequence, each term increases by a constant difference. In this case, the difference is 2. So, if I consider the sequence of 1/a_n, it's an arithmetic sequence starting from 1/a₁, which is 1, and each term increases by 2.Let me write that down. If I let b_n = 1/a_n, then the recurrence relation becomes b_{n+1} = b_n + 2. And since b₁ = 1/a₁ = 1, this is indeed an arithmetic sequence with first term 1 and common difference 2.So, the general formula for an arithmetic sequence is b_n = b₁ + (n - 1)d, where d is the common difference. Plugging in the values we have, b_n = 1 + (n - 1)*2. Simplifying that, b_n = 1 + 2n - 2, which simplifies further to b_n = 2n - 1.Therefore, since b_n = 1/a_n, we can write 1/a_n = 2n - 1. So, to find a_n, we just take the reciprocal of both sides. That gives us a_n = 1/(2n - 1).Now, the problem is asking for a_{20}. So, plugging n = 20 into the formula, we get a_{20} = 1/(2*20 - 1) = 1/(40 - 1) = 1/39.Wait, let me double-check that. If n = 20, then 2n is 40, minus 1 is 39, so yes, a_{20} is 1/39. That seems straightforward.Just to make sure I didn't make any mistakes, let me verify with the first few terms. We know a₁ is 1. Then, using the recurrence relation, 1/a₂ = 1/a₁ + 2 = 1 + 2 = 3, so a₂ = 1/3. Then, 1/a₃ = 1/a₂ + 2 = 3 + 2 = 5, so a₃ = 1/5. Similarly, a₄ = 1/7, and so on. So, the pattern is indeed a_n = 1/(2n - 1). Therefore, a_{20} = 1/39.I think that makes sense. The sequence of reciprocals is increasing by 2 each time, so the reciprocals are 1, 3, 5, 7, ..., which are the odd numbers. Hence, the nth term is 2n - 1, and taking the reciprocal gives us a_n = 1/(2n - 1). So, for the 20th term, it's 1/(2*20 - 1) = 1/39.Yeah, I'm pretty confident that's the right answer. I don't see any errors in my reasoning. It all checks out with the initial terms and the recurrence relation given.**Final Answer**The value of ( a_{20} ) is boxed{dfrac{1}{39}}.

answer：Okay, so I have this geometry problem here. It says: From point ( P ) on arc ( BC ) of the circumcircle of triangle ( ABC ), perpendiculars ( PX, PY, PZ ) are dropped to ( BC, CA ), and ( AB ) respectively. I need to prove that (frac{BC}{PX} = frac{AC}{PY} + frac{AB}{PZ}). Hmm, that's interesting.First, let me try to visualize the problem. Triangle ( ABC ) is inscribed in a circle, so it's a cyclic triangle. Point ( P ) is on the circumcircle, specifically on arc ( BC ). That means ( P ) is not on the same side of ( BC ) as ( A ); it's on the opposite arc. From ( P ), we drop perpendiculars to each side of the triangle, hitting ( BC ) at ( X ), ( CA ) at ( Y ), and ( AB ) at ( Z ).I remember that when you have a point on the circumcircle, certain properties hold. Maybe something related to cyclic quadrilaterals or inscribed angles? Also, the perpendiculars from ( P ) to the sides might form right angles, which could be useful.Let me think about the areas involved. Since ( PX, PY, PZ ) are perpendiculars, they represent the heights from ( P ) to each side. Maybe I can relate these heights to the areas of the triangles involved.Wait, the problem involves ratios of the sides to these perpendiculars. So, (frac{BC}{PX}), (frac{AC}{PY}), and (frac{AB}{PZ}). That makes me think of the formula for the area of a triangle, which is (frac{1}{2} times text{base} times text{height}). If I consider the areas of triangles ( PBC ), ( PCA ), and ( PAB ), their areas can be expressed in terms of these bases and heights.Let me denote the areas as follows:- Area of ( triangle PBC ) is ( frac{1}{2} times BC times PX ).- Area of ( triangle PCA ) is ( frac{1}{2} times AC times PY ).- Area of ( triangle PAB ) is ( frac{1}{2} times AB times PZ ).But how does this help me? I need to relate these areas in a way that connects ( frac{BC}{PX} ), ( frac{AC}{PY} ), and ( frac{AB}{PZ} ).Wait, since ( P ) is on the circumcircle, maybe there's a relation between these areas. I recall that in cyclic quadrilaterals, opposite angles sum to 180 degrees. But ( P ) is on the circumcircle, so ( ABCP ) is cyclic. That might mean that angles at ( P ) relate to angles in the triangle.Alternatively, maybe I can use trigonometric identities. Since ( P ) is on the circumcircle, the angles subtended by the sides from ( P ) might have specific relations.Let me think about the sine formula. In triangle ( ABC ), the sides are proportional to the sines of the opposite angles. Maybe something similar applies here.Wait, another thought: since ( P ) is on the circumcircle, the distances from ( P ) to the sides might relate to the sines of certain angles. Maybe the angles between the perpendiculars?Alternatively, perhaps I can use coordinate geometry. If I assign coordinates to the triangle and point ( P ), I might be able to express the perpendiculars in terms of coordinates and derive the required relation. But that might get complicated.Wait, going back to areas. If I consider the areas of the three smaller triangles ( PBC ), ( PCA ), and ( PAB ), their sum should be equal to the area of triangle ( ABC ). But I'm not sure if that's directly useful here.Wait, no. Actually, since ( P ) is on the circumcircle, the areas of ( PBC ), ( PCA ), and ( PAB ) might have some relation. Let me think.Alternatively, maybe I can use inversion or some projective geometry, but that might be overcomplicating things.Wait, another idea: since ( P ) is on the circumcircle, the power of point ( P ) with respect to the circumcircle is zero. But I'm not sure how that relates to the perpendiculars.Wait, perhaps I can use similar triangles. If I can find similar triangles involving ( PX, PY, PZ ), maybe I can relate the ratios.Alternatively, maybe trigonometric Ceva's theorem. Since ( P ) is on the circumcircle, the cevians from ( P ) might satisfy certain trigonometric relations.Wait, let me recall Ceva's theorem. It states that for concurrent cevians, the product of certain ratios equals 1. But in this case, the cevians are perpendiculars, so maybe not directly applicable.Wait, but since ( P ) is on the circumcircle, perhaps the cevians have some orthocentric properties.Alternatively, maybe I can use the fact that the product of the lengths of the perpendiculars relates to the sides.Wait, I'm getting a bit stuck. Let me try to write down what I know:- ( P ) is on the circumcircle of ( ABC ), so ( angle BPC = 180^circ - angle BAC ).- The perpendiculars from ( P ) to the sides are ( PX, PY, PZ ).- I need to prove ( frac{BC}{PX} = frac{AC}{PY} + frac{AB}{PZ} ).Hmm. Maybe I can express ( PX, PY, PZ ) in terms of the sides and some trigonometric functions.Let me consider the distances ( PX, PY, PZ ). Since they are perpendiculars, they can be expressed as:- ( PX = PB sin theta_1 ), where ( theta_1 ) is the angle between ( PB ) and ( BC ).Similarly,- ( PY = PC sin theta_2 ),- ( PZ = PA sin theta_3 ).But I'm not sure if this is helpful.Wait, another approach: use coordinates. Let me place triangle ( ABC ) in the coordinate plane, maybe with ( BC ) on the x-axis for simplicity.Let’s assume ( B ) is at ( (0, 0) ), ( C ) is at ( (a, 0) ), and ( A ) is somewhere in the plane, say ( (d, e) ). Then, the circumcircle can be determined, and point ( P ) can be parameterized on arc ( BC ).But this might involve a lot of calculations. Maybe it's better to use trigonometric identities.Wait, another idea: use areas again. If I denote the area of triangle ( ABC ) as ( S ), then the areas of ( PBC ), ( PCA ), and ( PAB ) can be related to ( S ).But I think I need a different approach. Maybe using vectors or complex numbers?Wait, perhaps I can use the fact that ( P ) lies on the circumcircle, so it can be represented as a complex number on the unit circle (if we normalize the circumradius to 1). Then, the perpendiculars can be expressed in terms of complex arguments.Hmm, this might get too involved, but let's try.Let me denote the circumradius as ( R ). Then, the sides can be expressed as ( BC = 2R sin A ), ( AC = 2R sin B ), ( AB = 2R sin C ).Wait, that's useful. So, ( BC = 2R sin A ), ( AC = 2R sin B ), ( AB = 2R sin C ).Now, the perpendiculars ( PX, PY, PZ ) can be related to the distances from ( P ) to the sides.In triangle ( ABC ), the distance from a point ( P ) to side ( BC ) is ( PX = 2R sin angle BPC sin A ). Wait, is that correct?Wait, no. Let me recall that in a triangle, the distance from a point on the circumcircle to a side can be expressed in terms of the sine of the angle at that point.Wait, actually, if ( P ) is on the circumcircle, then the distance from ( P ) to side ( BC ) is ( PX = 2R sin angle BPC sin A ). Wait, I'm not sure.Alternatively, maybe ( PX = R sin angle BPC ). Hmm.Wait, perhaps I can use the formula for the distance from a point to a line in terms of coordinates, but that might not be straightforward.Wait, another thought: in triangle ( ABC ), the distance from ( P ) to ( BC ) is ( PX = frac{2 times text{Area of } triangle PBC}{BC} ).Similarly, ( PY = frac{2 times text{Area of } triangle PCA}{AC} ), and ( PZ = frac{2 times text{Area of } triangle PAB}{AB} ).So, if I can express the areas of these triangles in terms of ( R ) and the angles, maybe I can relate them.Wait, since ( P ) is on the circumcircle, the areas of ( triangle PBC ), ( triangle PCA ), and ( triangle PAB ) can be expressed using the formula ( frac{1}{2} ab sin C ).So, for ( triangle PBC ), the area is ( frac{1}{2} PB times PC times sin angle BPC ).But ( angle BPC = 180^circ - angle BAC ), since ( P ) is on the circumcircle.Similarly, ( angle APC = 180^circ - angle ABC ), and ( angle APB = 180^circ - angle ACB ).So, the areas can be written as:- ( text{Area of } triangle PBC = frac{1}{2} PB times PC times sin (180^circ - A) = frac{1}{2} PB times PC times sin A ).Similarly,- ( text{Area of } triangle PCA = frac{1}{2} PC times PA times sin B ).- ( text{Area of } triangle PAB = frac{1}{2} PA times PB times sin C ).Now, since ( P ) is on the circumcircle, by the sine law, ( PA = 2R sin angle PBA ), but I'm not sure.Wait, actually, in the circumcircle, all points satisfy ( frac{a}{sin A} = 2R ). So, for point ( P ), the distances ( PA, PB, PC ) can be expressed in terms of the angles.Wait, let me recall that in a circle, the length of a chord is ( 2R sin theta ), where ( theta ) is half the angle subtended by the chord at the center.But since ( P ) is on the circumcircle, the angles subtended by the sides at ( P ) are related to the angles of the triangle.Wait, maybe I can express ( PB ) and ( PC ) in terms of ( R ) and the angles.Alternatively, perhaps I can use the fact that ( PB times PC = frac{BC^2}{4 sin^2 A} ) or something like that. Hmm, not sure.Wait, let's go back to the areas.We have:( text{Area of } triangle PBC = frac{1}{2} PB times PC times sin A ).Similarly,( text{Area of } triangle PCA = frac{1}{2} PC times PA times sin B ),( text{Area of } triangle PAB = frac{1}{2} PA times PB times sin C ).Now, the areas can also be expressed in terms of the heights:( text{Area of } triangle PBC = frac{1}{2} BC times PX ),( text{Area of } triangle PCA = frac{1}{2} AC times PY ),( text{Area of } triangle PAB = frac{1}{2} AB times PZ ).So, equating these two expressions for the areas:For ( triangle PBC ):( frac{1}{2} PB times PC times sin A = frac{1}{2} BC times PX ).Simplifying:( PB times PC times sin A = BC times PX ).Similarly,For ( triangle PCA ):( PC times PA times sin B = AC times PY ).For ( triangle PAB ):( PA times PB times sin C = AB times PZ ).So, we have:1. ( PB times PC times sin A = BC times PX ).2. ( PC times PA times sin B = AC times PY ).3. ( PA times PB times sin C = AB times PZ ).Now, let's solve each equation for ( PX, PY, PZ ):1. ( PX = frac{PB times PC times sin A}{BC} ).2. ( PY = frac{PC times PA times sin B}{AC} ).3. ( PZ = frac{PA times PB times sin C}{AB} ).Now, the goal is to prove that:( frac{BC}{PX} = frac{AC}{PY} + frac{AB}{PZ} ).Let me substitute the expressions for ( PX, PY, PZ ) into this equation.First, compute ( frac{BC}{PX} ):( frac{BC}{PX} = frac{BC}{frac{PB times PC times sin A}{BC}} = frac{BC^2}{PB times PC times sin A} ).Similarly,( frac{AC}{PY} = frac{AC}{frac{PC times PA times sin B}{AC}} = frac{AC^2}{PC times PA times sin B} ).( frac{AB}{PZ} = frac{AB}{frac{PA times PB times sin C}{AB}} = frac{AB^2}{PA times PB times sin C} ).So, we need to show:( frac{BC^2}{PB times PC times sin A} = frac{AC^2}{PC times PA times sin B} + frac{AB^2}{PA times PB times sin C} ).Hmm, that's a bit complicated. Let me see if I can manipulate this.First, let's note that in triangle ( ABC ), by the Law of Sines:( frac{BC}{sin A} = frac{AC}{sin B} = frac{AB}{sin C} = 2R ).So, ( BC = 2R sin A ), ( AC = 2R sin B ), ( AB = 2R sin C ).Let me substitute these into the equation.First, compute ( BC^2 = (2R sin A)^2 = 4R^2 sin^2 A ).Similarly,( AC^2 = 4R^2 sin^2 B ),( AB^2 = 4R^2 sin^2 C ).Now, substitute these into the equation:( frac{4R^2 sin^2 A}{PB times PC times sin A} = frac{4R^2 sin^2 B}{PC times PA times sin B} + frac{4R^2 sin^2 C}{PA times PB times sin C} ).Simplify each term:Left side:( frac{4R^2 sin^2 A}{PB times PC times sin A} = frac{4R^2 sin A}{PB times PC} ).First term on the right:( frac{4R^2 sin^2 B}{PC times PA times sin B} = frac{4R^2 sin B}{PC times PA} ).Second term on the right:( frac{4R^2 sin^2 C}{PA times PB times sin C} = frac{4R^2 sin C}{PA times PB} ).So, the equation becomes:( frac{4R^2 sin A}{PB times PC} = frac{4R^2 sin B}{PC times PA} + frac{4R^2 sin C}{PA times PB} ).We can factor out ( 4R^2 ) from all terms:( frac{sin A}{PB times PC} = frac{sin B}{PC times PA} + frac{sin C}{PA times PB} ).Now, let me denote ( PA = x ), ( PB = y ), ( PC = z ). So, the equation becomes:( frac{sin A}{y z} = frac{sin B}{z x} + frac{sin C}{x y} ).Multiply both sides by ( x y z ):( sin A times x = sin B times y + sin C times z ).So, we have:( x sin A = y sin B + z sin C ).Now, I need to show that this holds true given that ( P ) is on the circumcircle.Wait, in triangle ( ABC ), by the Law of Sines, we have ( frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} = 2R ).But how does this relate to ( PA, PB, PC )?Wait, since ( P ) is on the circumcircle, perhaps the distances ( PA, PB, PC ) can be related to the sides and angles.Wait, another thought: in triangle ( ABC ), the distances from ( P ) to the vertices can be expressed using the Law of Cosines in triangles ( PAB ), ( PBC ), ( PCA ).But I'm not sure.Wait, another idea: use the fact that in cyclic quadrilaterals, the product of the lengths can be related.Wait, actually, in triangle ( ABC ), point ( P ) is on the circumcircle, so by the Law of Sines in triangles ( PAB ), ( PBC ), ( PCA ), we have:In ( triangle PAB ):( frac{PA}{sin angle PBA} = 2R ).Similarly,In ( triangle PBC ):( frac{PB}{sin angle PCB} = 2R ).In ( triangle PCA ):( frac{PC}{sin angle PAC} = 2R ).But I'm not sure if this helps.Wait, perhaps I can use the fact that angles subtended by the same chord are equal.Wait, since ( P ) is on the circumcircle, ( angle PAB = angle PCB ), because they subtend the same arc ( PB ).Similarly, ( angle PBA = angle PCA ), and ( angle PCA = angle PBA ).Wait, maybe I can relate the angles in such a way that the sine terms can be connected.Alternatively, perhaps I can use the fact that in triangle ( ABC ), the sum of the angles is ( 180^circ ), so ( A + B + C = 180^circ ).But I'm not sure.Wait, going back to the equation we have:( x sin A = y sin B + z sin C ).Where ( x = PA ), ( y = PB ), ( z = PC ).Is there a known relation in triangle geometry that connects these?Wait, I recall that in a triangle, if a point lies on the circumcircle, then the distances from that point to the vertices satisfy certain relations. Maybe something like ( PA sin A = PB sin B + PC sin C ). Wait, that's exactly what we have here!Yes, in fact, there is a theorem that states that for a point ( P ) on the circumcircle of triangle ( ABC ), the following holds:( PA sin A = PB sin B + PC sin C ).So, this is exactly the equation we derived. Therefore, the equation ( x sin A = y sin B + z sin C ) holds true, which means our earlier steps are correct.Thus, substituting back, we have:( frac{BC}{PX} = frac{AC}{PY} + frac{AB}{PZ} ).Therefore, the proof is complete.

answer：Okay, so I need to find the domain of the function ( g(x) = sqrt{x + 3} + log_{2}(6 - x) ). Hmm, the domain of a function refers to all the possible input values (x-values) for which the function is defined. Since this function has two parts—a square root and a logarithm—I need to make sure both parts are defined for the same x-values.First, let me think about the square root part: ( sqrt{x + 3} ). For a square root to be defined, the expression inside the square root (the radicand) must be greater than or equal to zero. So, I can set up the inequality:( x + 3 geq 0 )Solving for x, I subtract 3 from both sides:( x geq -3 )Okay, so x has to be greater than or equal to -3 for the square root to be defined.Next, let's look at the logarithmic part: ( log_{2}(6 - x) ). For a logarithm to be defined, the argument (the expression inside the log) must be positive. So, I set up the inequality:( 6 - x > 0 )Solving for x, I subtract 6 from both sides:( -x > -6 )Then, I multiply both sides by -1. But wait, when I multiply or divide both sides of an inequality by a negative number, I have to reverse the inequality sign. So:( x < 6 )Alright, so x has to be less than 6 for the logarithm to be defined.Now, to find the domain of the entire function ( g(x) ), I need to find the values of x that satisfy both conditions simultaneously. That means x has to be greater than or equal to -3 and less than 6.So, combining these two inequalities:( -3 leq x < 6 )Let me double-check to make sure I didn't make any mistakes. For the square root, if x is -3, then ( sqrt{-3 + 3} = sqrt{0} = 0 ), which is defined. If x is just less than 6, say 5.9, then ( sqrt{5.9 + 3} = sqrt{8.9} ) is defined, and ( log_{2}(6 - 5.9) = log_{2}(0.1) ) is defined because 0.1 is positive. If x is exactly 6, then ( log_{2}(6 - 6) = log_{2}(0) ) is undefined, so x cannot be 6. Similarly, if x is less than -3, say -4, then ( sqrt{-4 + 3} = sqrt{-1} ) is undefined in real numbers. So, my domain seems correct.Looking back at the options provided:A: ( {x | x > 6} ) – This can't be right because x has to be less than 6.B: ( {x | -3 < x leq 6} ) – Close, but x can be equal to -3, and it shouldn't include 6.C: ( {x | x > -3} ) – This doesn't account for the upper limit of 6.D: ( {x | -3 leq x < 6} ) – This matches my solution.So, the correct answer should be D.

answer：First, I need to determine the total number of possible outcomes when rolling a 12-sided die. Since the die has 12 faces, there are 12 possible outcomes.Next, I identify the perfect squares that are possible outcomes. A perfect square is a number that can be expressed as the square of an integer. The perfect squares less than or equal to 12 are 1 (1²), 4 (2²), and 9 (3²). Therefore, there are 3 successful outcomes.Finally, I calculate the probability by dividing the number of successful outcomes by the total number of possible outcomes. This gives me 3 divided by 12, which simplifies to 1/4.