answer：To compute (sin 30^circ), I start by considering a 30-60-90 triangle, which is a special right triangle with angles of 30°, 60°, and 90°. In such a triangle, the sides are in the ratio 1 : (sqrt{3}) : 2, where the side opposite the 30° angle is the shortest side.Since the hypotenuse of the triangle is 2 units, the side opposite the 30° angle, which corresponds to the sine of 30°, is half the hypotenuse. Therefore, (sin 30^circ = frac{1}{2}).

answer：Alright, so I've got this problem here with a regular octagon, and I need to find its area. The diagram shows a square BDEF, and it's given that AB equals BC equals 1. Hmm, okay, let me try to visualize this. First, I remember that a regular octagon has all sides equal and all internal angles equal. But in this case, the octagon is drawn inside a square, which might mean that some of its sides are aligned with the square's sides or maybe at an angle. The Asymptote code is provided, but I think I can work this out without needing to parse the code in detail.Given that AB and BC are both 1, and looking at the labels on the diagram, I think triangle ABC is a right triangle because points A, B, and C are connected in a way that suggests a right angle at B. Since AB and BC are both 1, this must be a 45-45-90 triangle. In such triangles, the hypotenuse is √2 times the length of each leg. So, AC should be √2. That might be useful later.Now, the square BDEF is mentioned. Since it's a square, all its sides are equal, and all its angles are right angles. Points B, D, E, and F are the vertices of this square. Looking at the Asymptote code, it seems like B is at (-x, 2+2x), D is at (2+x, 2+2x), E is at (2+x, 0), and F is at (-x, 0). Hmm, that's a bit confusing, but maybe I can figure out the side length of the square.If I consider the coordinates, the distance between B and D should be the same as the distance between D and E, and so on. Let's calculate the distance between B and D. The x-coordinate changes from -x to 2+x, so that's a difference of 2 + 2x. The y-coordinate remains the same at 2+2x, so the distance between B and D is just 2 + 2x. Similarly, the distance between D and E is from (2+x, 2+2x) to (2+x, 0), which is a vertical distance of 2+2x. So, the side length of square BDEF is 2 + 2x.Wait, but x is defined as sqrt(2) in the Asymptote code. So, substituting x with sqrt(2), the side length becomes 2 + 2*sqrt(2). Therefore, the area of square BDEF is (2 + 2*sqrt(2)) squared. Let me compute that:(2 + 2√2)^2 = 2^2 + 2*2*2√2 + (2√2)^2 = 4 + 8√2 + 8 = 12 + 8√2.Hmm, but I'm not sure if I need the area of the square or the octagon. The problem asks for the area of the regular octagon, so maybe the square is just a part of the figure.Looking back at the diagram, the octagon is drawn inside the square, and there are triangles attached to each side of the square. Specifically, there are four right isosceles triangles, each with legs of length 1, attached to the square. These triangles are probably the ones with sides AB and BC equal to 1.Since each of these triangles has legs of length 1, their area is (1*1)/2 = 0.5 each. There are four of them, so the total area of the triangles is 4*0.5 = 2.If I subtract the area of these triangles from the area of the square, I should get the area of the octagon. So, the area of the octagon would be the area of the square minus the area of the four triangles:Area of octagon = (12 + 8√2) - 2 = 10 + 8√2.Wait, but that doesn't seem right because I think the side length of the octagon is related to the triangles. Earlier, I found that AC is √2, which might be the side length of the octagon. If the side length of the octagon is √2, then maybe I can use the formula for the area of a regular octagon.The formula for the area of a regular octagon with side length 'a' is 2(1 + √2)a². So, plugging in a = √2:Area = 2(1 + √2)*(√2)² = 2(1 + √2)*2 = 4(1 + √2) = 4 + 4√2.Hmm, that's different from what I got earlier. I think I made a mistake in calculating the area of the square. Let me double-check.The side length of square BDEF was calculated as 2 + 2x, where x = sqrt(2). So, 2 + 2*sqrt(2). Squaring that:(2 + 2√2)^2 = 4 + 8√2 + 8 = 12 + 8√2. That seems correct.But if the octagon's area is 4 + 4√2, and the square's area is 12 + 8√2, subtracting the triangles' area (2) gives 10 + 8√2, which doesn't match. So, maybe my initial approach is wrong.Perhaps instead of subtracting the triangles from the square, I should consider that the octagon is formed by the square minus the four triangles. But if the octagon is regular, all its sides and angles are equal, so maybe the triangles are part of the octagon's sides.Wait, looking back, the octagon is drawn with points A, B, C, D, E, F, G, H. So, it's a regular octagon inscribed in the square BDEF, with four triangles cut off from the square's corners. Each triangle has legs of length 1, so their hypotenuses are √2, which would be the sides of the octagon.Therefore, the side length of the octagon is √2, and the area can be calculated using the regular octagon area formula: 2(1 + √2)a².Plugging in a = √2:Area = 2(1 + √2)*(√2)² = 2(1 + √2)*2 = 4(1 + √2) = 4 + 4√2.Yes, that makes sense. So, the area of the regular octagon is 4 + 4√2 square units.

answer：Alright, so I have this problem about linear regression and correlation coefficients. It's about a business that's been collecting data on their monthly advertising investment and monthly sales volume over seven months. They want to use a linear regression model to fit the relationship between advertising investment (x) and sales volume (y). There are two parts: first, explaining why a linear regression model is suitable using the correlation coefficient, and second, finding the regression equation and predicting the advertising investment needed to exceed 70,000 pieces in sales.Okay, let's start with part one. I need to explain using the correlation coefficient why a linear regression model is appropriate. From what I remember, the correlation coefficient measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate a strong linear relationship, and values close to 0 indicate a weak or no linear relationship.So, if the correlation coefficient is high, say close to 1, that would mean that as advertising investment increases, sales volume also increases in a predictable, linear way. That would make a linear regression model a good fit because the relationship is strong and linear.But wait, the problem doesn't give me the correlation coefficient directly. It gives me some reference data: the sum of the products of the deviations for x and y is 150, and the sum of the squared deviations for y is 820. Also, it mentions that the square root of 1435 is approximately 37.88. Hmm, okay.I think I need to calculate the correlation coefficient using the formula provided:r = [sum((x_i - x̄)(y_i - ȳ))] / [sqrt(sum((x_i - x̄)^2) * sum((y_i - ȳ)^2))]They've given me sum((x_i - x̄)(y_i - ȳ)) = 150, and sum((y_i - ȳ)^2) = 820. But I don't have sum((x_i - x̄)^2). Maybe I can calculate that.First, I need to find the mean of x and y. Let's see, the x values are 1, 2, 3, 4, 5, 6, 7. So, the mean of x, x̄, is (1+2+3+4+5+6+7)/7. That's 28/7 = 4.For y, the values are 28, 32, 35, 45, 49, 52, 60. So, ȳ is (28+32+35+45+49+52+60)/7. Let's add those up: 28+32=60, 60+35=95, 95+45=140, 140+49=189, 189+52=241, 241+60=301. So, 301/7 = 43.Okay, so x̄ is 4 and ȳ is 43.Now, to find sum((x_i - x̄)^2). Since x is 1,2,3,4,5,6,7, subtracting the mean (4) from each gives -3, -2, -1, 0, 1, 2, 3. Squaring each of these gives 9, 4, 1, 0, 1, 4, 9. Adding those up: 9+4=13, 13+1=14, 14+0=14, 14+1=15, 15+4=19, 19+9=28. So, sum((x_i - x̄)^2) is 28.Great, now I have all the components to calculate r.r = 150 / [sqrt(28 * 820)]Let me compute the denominator. First, 28 * 820. Let's see, 28*800=22400, and 28*20=560, so total is 22400 + 560 = 22960. So sqrt(22960). Hmm, that's a big number. Wait, but the problem mentions sqrt(1435) ≈ 37.88. Maybe I can factor 22960 to see if it relates to 1435.22960 divided by 16 is 1435, because 1435 * 16 = 22960. So sqrt(22960) = sqrt(16 * 1435) = 4 * sqrt(1435). And sqrt(1435) is approximately 37.88, so 4 * 37.88 = 151.52.So, the denominator is approximately 151.52.Therefore, r ≈ 150 / 151.52 ≈ 0.99.Wow, that's a very high correlation coefficient, almost 1. So, that means there's a very strong positive linear relationship between advertising investment and sales volume. Hence, a linear regression model is very suitable for fitting this relationship.Alright, that takes care of part one. Now, moving on to part two: finding the linear regression equation and predicting the advertising investment needed to exceed 70,000 pieces in sales.The regression line is given by ŷ = b̂x + â, where b̂ is the slope and â is the intercept. The formula for b̂ is sum((x_i - x̄)(y_i - ȳ)) / sum((x_i - x̄)^2). We already have these values: sum((x_i - x̄)(y_i - ȳ)) = 150 and sum((x_i - x̄)^2) = 28. So, b̂ = 150 / 28.Let me compute that: 150 divided by 28. 28*5=140, so 150-140=10. So, 5 and 10/28, which simplifies to 5 and 5/14, or as an improper fraction, (5*14 +5)/14 = 75/14. So, b̂ = 75/14.Next, to find â, the intercept. The formula is â = ȳ - b̂x̄. We have ȳ = 43, x̄ = 4, and b̂ = 75/14.So, â = 43 - (75/14)*4. Let's compute that: (75/14)*4 = 300/14 = 150/7 ≈ 21.4286.So, â = 43 - 150/7. 43 is 301/7, so 301/7 - 150/7 = 151/7 ≈ 21.5714.Therefore, the regression equation is ŷ = (75/14)x + 151/7.Now, to predict the amount of monthly advertising investment when the monthly sales volume can exceed 70,000 pieces. Wait, the sales volume y is in units of ten thousand pieces. So, 70,000 pieces is 7 in these units.So, we need to find x such that ŷ > 7.So, set up the inequality:(75/14)x + 151/7 > 7First, let's convert 7 to sevenths to make it easier. 7 = 49/7.So, (75/14)x + 151/7 > 49/7Subtract 151/7 from both sides:(75/14)x > (49/7 - 151/7) = (-102)/7Wait, that can't be right. If I subtract 151/7 from both sides, I get:(75/14)x > 49/7 - 151/7 = (-102)/7But that would mean x is negative, which doesn't make sense because advertising investment can't be negative. Did I make a mistake?Wait, no. Let me double-check. The sales volume y is in ten thousand pieces, so 70,000 pieces is 7 ten thousand pieces. So, we need ŷ > 7.But our regression equation is ŷ = (75/14)x + 151/7.So, set (75/14)x + 151/7 > 7.Convert 7 to sevenths: 7 = 49/7.So, (75/14)x + 151/7 > 49/7Subtract 151/7 from both sides:(75/14)x > (49/7 - 151/7) = (-102)/7Hmm, that's negative. That doesn't make sense because x is positive. Did I set up the inequality correctly?Wait, maybe I misinterpreted the units. The sales volume y is in ten thousand pieces, so 70,000 pieces is 70,000 / 10,000 = 7. So, y needs to be greater than 7.But according to the regression equation, when x increases, y increases. So, to get y > 7, we need to find x such that (75/14)x + 151/7 > 7.Wait, but when x is 0, y is 151/7 ≈ 21.57, which is already greater than 7. That can't be right because in the data, when x=1, y=28, which is 2.8 in ten thousand pieces? Wait, no, hold on. Wait, the units are ten thousand pieces. So, y=28 means 28 ten thousand pieces, which is 280,000 pieces. Wait, that contradicts the previous statement.Wait, hold on. Let me re-examine the problem.The table says:Monthly Advertising Investment x (ten thousand yuan): 1,2,3,4,5,6,7Monthly Sales Volume y (ten thousand pieces): 28,32,35,45,49,52,60So, y is in ten thousand pieces, so 28 is 280,000 pieces, not 28,000. So, 70,000 pieces is 7 ten thousand pieces, which is 7.But in the data, when x=7, y=60, which is 600,000 pieces. So, to get y=7 (70,000 pieces), we need to find x such that ŷ=7.But wait, in the data, the smallest y is 28, which is 280,000 pieces. So, 70,000 pieces is much lower than the observed data. That seems odd. Maybe I misread the units.Wait, the problem says "monthly sales volume y (in units of ten thousand pieces)". So, y=28 is 28 ten thousand pieces, which is 280,000 pieces. So, 70,000 pieces is 7 ten thousand pieces, which is y=7.But in the data, y starts at 28, which is much higher than 7. So, to get y=7, which is lower than the observed minimum y, we would need to extrapolate the regression line below the observed data. That's not advisable because the relationship might not hold outside the observed range.But the problem asks to predict the amount of monthly advertising investment when the monthly sales volume can exceed 70,000 pieces, which is y=7. So, even though it's below the observed data, let's proceed with the calculation.So, set ŷ = 7:(75/14)x + 151/7 = 7Multiply both sides by 14 to eliminate denominators:75x + 302 = 98Wait, 151/7 *14 = 151*2 = 302, and 7*14=98.So, 75x + 302 = 98Subtract 302 from both sides:75x = 98 - 302 = -204So, x = -204 / 75 ≈ -2.72But x represents advertising investment in ten thousand yuan, and it can't be negative. So, this suggests that according to the regression model, to achieve a sales volume of 70,000 pieces, you would need a negative advertising investment, which doesn't make practical sense.This indicates that the regression model isn't suitable for predicting sales volumes below the observed minimum, as it leads to nonsensical results. Therefore, it's not possible to achieve a sales volume of 70,000 pieces with positive advertising investment based on this model.But the problem asks to predict the amount of monthly advertising investment when the monthly sales volume can exceed 70,000 pieces. Maybe I misinterpreted the units again. Let me check.Wait, y is in ten thousand pieces, so 70,000 pieces is 7 ten thousand pieces. But in the data, the smallest y is 28, which is 280,000 pieces. So, 70,000 pieces is much lower than the observed data. Therefore, the regression model can't reliably predict y values below 28 because it's extrapolation beyond the data range.Hence, it's not feasible to predict a positive x that would result in y=7. The model suggests that to get y=7, x would have to be negative, which isn't practical. Therefore, the business cannot achieve a sales volume of 70,000 pieces with positive advertising investment based on this model.But the problem still asks to predict it. Maybe I made a mistake in setting up the equation. Let me double-check.Wait, the regression equation is ŷ = (75/14)x + 151/7. So, to find x when ŷ > 70,000 pieces, which is y=7.Wait, but in the data, y is already much higher than 7. So, perhaps the problem meant 70,000 pieces, which is 7 ten thousand pieces, but in the data, y is 28,32,... So, maybe the problem has a typo, and they meant 70 ten thousand pieces, which is 700,000 pieces. That would make more sense.If that's the case, then y=70. So, set ŷ = 70.So, (75/14)x + 151/7 = 70Multiply both sides by 14:75x + 302 = 980Subtract 302:75x = 678x = 678 / 75 = 9.04So, x ≈ 9.04 ten thousand yuan, which is 90,400 yuan.That makes more sense. So, the business needs to invest approximately 9.04 ten thousand yuan in advertising to expect sales to exceed 70 ten thousand pieces, which is 700,000 pieces.But the problem specifically says "exceed 70,000 pieces", which is 7 ten thousand pieces. So, unless there's a typo, the answer would be that it's not possible with positive advertising investment. But since the problem asks to predict it, perhaps they intended 70 ten thousand pieces.Alternatively, maybe I misread the units. Let me check again.The problem states: "monthly sales volume y (in units of ten thousand pieces)". So, y=28 is 280,000 pieces. Therefore, 70,000 pieces is y=7, which is below the observed data. So, to get y=7, x would have to be negative, which isn't practical.Therefore, the answer is that it's not possible to achieve a sales volume of 70,000 pieces with positive advertising investment based on this model.But the problem says "predict the amount of monthly advertising investment when the monthly sales volume can exceed 70,000 pieces." So, maybe they expect us to proceed with the calculation despite it being below the observed data.In that case, solving for x when ŷ=7:(75/14)x + 151/7 = 7Multiply both sides by 14:75x + 302 = 9875x = 98 - 302 = -204x = -204 / 75 ≈ -2.72So, x ≈ -2.72 ten thousand yuan. But negative advertising investment doesn't make sense. Therefore, the business cannot achieve a sales volume of 70,000 pieces with positive advertising investment based on this model.But perhaps the problem expects us to ignore the practicality and just provide the mathematical answer. So, x ≈ -2.72, but since it's negative, it's not feasible.Alternatively, maybe I made a mistake in calculating the regression equation. Let me double-check.We have:b̂ = 150 / 28 = 75/14 ≈ 5.357â = ȳ - b̂x̄ = 43 - (75/14)*4 = 43 - (300/14) = 43 - 21.4286 ≈ 21.5714So, the regression equation is ŷ = 5.357x + 21.5714If we set ŷ = 7:5.357x + 21.5714 = 75.357x = 7 - 21.5714 = -14.5714x = -14.5714 / 5.357 ≈ -2.72Same result. So, it's consistent.Therefore, the conclusion is that to achieve a sales volume of 70,000 pieces, the business would need to invest approximately -2.72 ten thousand yuan, which is not possible. Hence, it's not feasible with positive advertising investment.But the problem asks to predict it, so maybe they expect us to state that it's not possible or to provide the negative value. Alternatively, perhaps I misread the units.Wait, another possibility: maybe y is in thousands of pieces, not ten thousand. Let me check the problem again.It says: "monthly sales volume y (in units of ten thousand pieces)". So, y=28 is 280,000 pieces. So, 70,000 pieces is y=7. So, it's correct.Therefore, the answer is that it's not possible to achieve a sales volume of 70,000 pieces with positive advertising investment based on this model.But the problem also says "can exceed 70,000 pieces". So, maybe they mean to find the x when y exceeds 70,000 pieces, which is y=7. But as we saw, that requires negative x.Alternatively, maybe they meant 70 ten thousand pieces, which is 700,000 pieces, y=70. Then, solving for x:ŷ = 7075/14 x + 151/7 = 70Multiply by 14:75x + 302 = 98075x = 678x = 678 / 75 = 9.04So, x ≈ 9.04 ten thousand yuan.That makes sense. So, perhaps the problem had a typo, and they meant 70 ten thousand pieces instead of 70,000 pieces. In that case, the answer would be x ≈ 9.04.But since the problem specifically says 70,000 pieces, which is y=7, and that leads to a negative x, which is not feasible, I think the correct answer is that it's not possible with positive advertising investment.However, since the problem asks to predict it, maybe they expect us to provide the mathematical answer regardless of feasibility. So, x ≈ -2.72 ten thousand yuan.But that's negative, so perhaps we should state that it's not possible.Alternatively, maybe I made a mistake in interpreting the units. Let me check again.Wait, the problem says "monthly sales volume y (in units of ten thousand pieces)". So, y=28 is 280,000 pieces. Therefore, 70,000 pieces is y=7, which is below the observed data. So, to get y=7, x would have to be negative.Therefore, the business cannot achieve a sales volume of 70,000 pieces with positive advertising investment based on this model.But the problem asks to predict it, so maybe they expect us to state that it's not possible.Alternatively, perhaps I should consider that the regression model is only valid within the range of the data, and extrapolating beyond that is unreliable. Therefore, it's not possible to predict a sales volume below the observed minimum with this model.In conclusion, based on the linear regression model, to achieve a sales volume of 70,000 pieces, the required advertising investment would be negative, which is not feasible. Therefore, it's not possible to predict a positive advertising investment that would result in a sales volume of 70,000 pieces using this model.

answer：First, I need to determine the total amount of flour Sarah purchased. She bought 5 pounds of rye flour, 10 pounds of whole-wheat bread flour, and 3 pounds of chickpea flour. Adding these together:5 pounds (rye) + 10 pounds (whole-wheat bread) + 3 pounds (chickpea) = 18 pounds.Next, I should account for the flour she already had at home. Sarah had 2 pounds of whole-wheat pastry flour. Adding this to the purchased flour:18 pounds (purchased) + 2 pounds (existing) = 20 pounds.Therefore, Sarah now has a total of 20 pounds of flour.

answer：First, I observe that both terms in the expression ( x^3 - 16x ) have a common factor of ( x ). I'll factor out the ( x ) to simplify the expression.After factoring out ( x ), the expression becomes ( x(x^2 - 16) ). Now, I notice that ( x^2 - 16 ) is a difference of squares, since ( 16 ) is ( 4^2 ).Using the difference of squares formula, ( a^2 - b^2 = (a + b)(a - b) ), I can factor ( x^2 - 16 ) into ( (x + 4)(x - 4) ).Finally, combining all the factors, the fully factorized form of the original expression is ( x(x + 4)(x - 4) ).

answer：Okay, so I have this function f(x) = x - a/x, and it's given that f(1) = 2. I need to figure out two things: first, whether the function is even, odd, or neither, and second, whether it's increasing or decreasing on the interval (1, +∞). Let me start with part (I).Alright, for part (I), determining the parity of the function. I remember that a function is even if f(-x) = f(x) for all x in its domain, and it's odd if f(-x) = -f(x) for all x in its domain. If neither of these conditions hold, then the function is neither even nor odd.But before I can check the parity, I need to find the value of 'a' because the function is given in terms of 'a', and I know that f(1) = 2. So, let me substitute x = 1 into the function:f(1) = 1 - a/1 = 1 - a.And it's given that f(1) = 2, so:1 - a = 2.Solving for 'a', I subtract 1 from both sides:-a = 1,then multiply both sides by -1:a = -1.So, the function becomes f(x) = x - (-1)/x = x + 1/x.Alright, now that I have the function f(x) = x + 1/x, I can check its parity. Let's compute f(-x):f(-x) = (-x) + 1/(-x) = -x - 1/x.Now, let's see if this is equal to f(x) or -f(x). The original function f(x) is x + 1/x. So, f(-x) = -x - 1/x, which is equal to -(x + 1/x) = -f(x). Therefore, f(-x) = -f(x), which means the function is odd. That answers part (I).Moving on to part (II), I need to determine the monotonicity of f(x) on the interval (1, +∞). Monotonicity refers to whether the function is increasing or decreasing over that interval. To figure this out, I can use calculus—specifically, the first derivative test.First, let me find the derivative of f(x). The function is f(x) = x + 1/x. The derivative of x with respect to x is 1, and the derivative of 1/x with respect to x is -1/x². So, putting it together:f'(x) = 1 - 1/x².Now, to determine where the function is increasing or decreasing, I need to analyze the sign of f'(x). If f'(x) is positive on an interval, the function is increasing there; if it's negative, the function is decreasing.Let's analyze f'(x) = 1 - 1/x². I can rewrite this as:f'(x) = (x² - 1)/x².So, f'(x) = (x² - 1)/x².Now, since x is in the interval (1, +∞), x is positive, and x² is also positive. Therefore, the denominator x² is always positive. The sign of f'(x) then depends on the numerator, which is x² - 1.Let's see when x² - 1 is positive. Since x > 1, x² > 1, so x² - 1 > 0. Therefore, the numerator is positive, and since the denominator is always positive, f'(x) is positive for all x in (1, +∞).Since the derivative f'(x) is positive on (1, +∞), the function f(x) is increasing on that interval.Alternatively, if I didn't want to use calculus, I could consider the difference f(x2) - f(x1) for x2 > x1 > 1 and show that it's positive, which would imply the function is increasing. Let me try that approach as a verification.Let x2 > x1 > 1. Then:f(x2) - f(x1) = (x2 + 1/x2) - (x1 + 1/x1) = (x2 - x1) + (1/x2 - 1/x1).Simplify the second term:1/x2 - 1/x1 = (x1 - x2)/(x1 x2).So, f(x2) - f(x1) = (x2 - x1) + (x1 - x2)/(x1 x2) = (x2 - x1)(1 - 1/(x1 x2)).Since x2 > x1, x2 - x1 > 0. Also, since x1 > 1 and x2 > 1, x1 x2 > 1, so 1/(x1 x2) < 1, which means 1 - 1/(x1 x2) > 0. Therefore, both factors (x2 - x1) and (1 - 1/(x1 x2)) are positive, so their product is positive. Hence, f(x2) - f(x1) > 0, which implies f(x2) > f(x1). Therefore, the function is increasing on (1, +∞).Both methods confirm that the function is increasing on the interval (1, +∞).Wait, just to make sure I didn't make any mistakes in my reasoning. Let me double-check the derivative approach. f'(x) = 1 - 1/x². For x > 1, x² > 1, so 1/x² < 1, which means 1 - 1/x² > 0. So, f'(x) is indeed positive, meaning the function is increasing. That seems correct.And in the difference approach, I had f(x2) - f(x1) = (x2 - x1)(1 - 1/(x1 x2)). Since x2 > x1 > 1, both terms are positive, so the product is positive. That also seems correct.I think I'm confident with both methods. So, the function is increasing on (1, +∞).**Final Answer**Ⅰ. The function is boxed{text{odd}}.Ⅱ. The function is boxed{text{increasing}} on the interval ((1, +infty)).