answer：Okay, so I have this problem where I'm given that f'(x) is the derivative of f(x), and f'(x) is equal to (a-1)x² + a x + 1. It also says that f'(x) is an even function. I need to find the increasing interval of f(x). Hmm, okay, let's break this down step by step.First, I remember that an even function satisfies the condition f'(-x) = f'(x) for all x. So, if I plug in -x into f'(x), it should be the same as f'(x). Let me write that out:f'(-x) = (a - 1)(-x)² + a(-x) + 1.Simplifying that, (-x)² is just x², so it becomes:f'(-x) = (a - 1)x² - a x + 1.Now, since f'(x) is even, this should equal f'(x), which is (a - 1)x² + a x + 1. So, setting them equal:(a - 1)x² - a x + 1 = (a - 1)x² + a x + 1.Hmm, okay, let's subtract (a - 1)x² and 1 from both sides to simplify:- a x = a x.So, that simplifies to -a x = a x. If I bring all terms to one side:- a x - a x = 0 -2 a x = 0.This equation needs to hold for all x, right? Because f'(x) is even for all x. The only way this can be true for all x is if the coefficient of x is zero. So, -2 a = 0, which implies that a = 0.Okay, so a is 0. Now, let's substitute a = 0 back into f'(x):f'(x) = (0 - 1)x² + 0 x + 1 f'(x) = -x² + 1.Alright, so f'(x) is -x² + 1. Now, I need to find the intervals where f(x) is increasing. Since f'(x) is the derivative, f(x) is increasing where f'(x) > 0.So, let's solve the inequality:- x² + 1 > 0.Let me rearrange that:- x² + 1 > 0 1 - x² > 0 x² < 1.So, x² < 1 implies that x is between -1 and 1. Therefore, the solution is -1 < x < 1.So, the function f(x) is increasing on the interval (-1, 1).Wait, let me double-check my steps to make sure I didn't make a mistake. First, I used the property of even functions correctly by substituting -x into f'(x). Then, I set f'(-x) equal to f'(x) and simplified, which led me to find that a must be 0. Substituting back, I got f'(x) = -x² + 1. Then, solving for where f'(x) is positive, I found x between -1 and 1. That seems correct.Just to be thorough, let me think about the graph of f'(x). Since f'(x) is a quadratic function opening downward (because the coefficient of x² is negative), it has a maximum at its vertex. The roots of f'(x) = 0 are at x = 1 and x = -1, which are the points where the function changes from positive to negative. So, between -1 and 1, the function is positive, meaning f(x) is increasing there, and outside of that interval, it's decreasing. That makes sense.So, I think I've got it right. The increasing interval is (-1, 1).**Final Answer**The increasing interval of ( f(x) ) is boxed{(-1, 1)}.

answer：Alright, so I'm trying to figure out how many water bottles Ellis and her family will need for their road trip. Let me break this down step by step to make sure I understand everything correctly.First, there are four people in total: Ellis, her mother, her sister, and her aunt. That's four individuals. They're going to be traveling to their destination, which will take 8 hours, and then they'll drive back home, which will also take 8 hours. So, in total, they'll be on the road for 16 hours.Now, each person wants to drink half a bottle of water every hour. That means for each hour they're driving, each person will consume 0.5 bottles of water. Since there are four people, I need to calculate how much water they'll all drink together in one hour, and then multiply that by the total number of hours they'll be on the road.Let me write this out:1. **Calculate the water consumption per hour for all four people:** - Each person drinks 0.5 bottles per hour. - There are 4 people. - So, total water consumption per hour = 0.5 bottles/person * 4 people = 2 bottles per hour.2. **Calculate the total water consumption for the entire trip:** - They're driving for 8 hours to the destination and 8 hours back, so total driving time = 8 + 8 = 16 hours. - Total water consumption = 2 bottles/hour * 16 hours = 32 bottles.Wait, that seems straightforward, but let me double-check to make sure I haven't missed anything.- **Number of people:** 4 (Ellis, her mother, her sister, her aunt) – correct.- **Driving time:** 8 hours each way, so 16 hours total – correct.- **Water consumption per person per hour:** 0.5 bottles – correct.- **Total consumption per hour for all:** 0.5 * 4 = 2 bottles – correct.- **Total consumption for 16 hours:** 2 * 16 = 32 bottles – correct.Hmm, everything seems to add up. I don't think I've missed any details here. It looks like they'll need 32 bottles of water for the entire trip.Just to think about it another way, if each person drinks half a bottle every hour, then over 16 hours, each person would drink 0.5 * 16 = 8 bottles. With four people, that's 8 * 4 = 32 bottles. Yep, that matches my earlier calculation.So, I'm pretty confident that the total number of water bottles needed is 32.

answer：First, I need to subtract the two octal numbers (453_8) and (267_8). To do this, I'll align the numbers by their place values.Starting from the rightmost digit, I'll subtract the units place: (3_8 - 7_8). Since (3) is less than (7), I'll need to borrow (1) from the next higher place, making it (13_8 - 7_8 = 4_8).Next, I'll subtract the middle digits: (4_8 - 6_8). Again, (4) is less than (6), so I'll borrow (1) from the next higher place, resulting in (14_8 - 6_8 = 6_8).Finally, I'll subtract the leftmost digits: (3_8 - 2_8 = 1_8).Combining these results, the subtraction (453_8 - 267_8) equals (164_8).

answer：Okay, so I have this problem where I need to find the area of a triangle with vertices at (3, -3), (8, 4), and (3, 4). Hmm, let me think about how to approach this. I remember that in coordinate geometry, there are a few ways to find the area of a triangle when you know the coordinates of its vertices. One common method is using the shoelace formula, which I think involves multiplying coordinates in a certain way and then taking half the absolute difference. But I'm not entirely sure how to apply it correctly. Maybe I should try plotting the points first to visualize the triangle.Alright, let's plot the points. The first point is (3, -3), which is somewhere in the fourth quadrant. The second point is (8, 4), which is in the first quadrant, and the third point is (3, 4), also in the first quadrant. So, connecting these points, I can see that two of the points share the same x-coordinate, which is 3. That means the line between (3, -3) and (3, 4) is vertical. Similarly, the line between (3, 4) and (8, 4) is horizontal because they share the same y-coordinate, 4. So, this triangle has a vertical side and a horizontal side, which makes me think it might be a right triangle. If that's the case, finding the area should be straightforward since the area of a right triangle is just half the product of the lengths of the two legs.Let me verify if it's a right triangle. In a right triangle, the sides should satisfy the Pythagorean theorem, which states that the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). First, I need to find the lengths of all three sides.Starting with the vertical side between (3, -3) and (3, 4). Since the x-coordinates are the same, the distance between these two points is just the difference in the y-coordinates. So, 4 - (-3) equals 7 units. That's one leg of the triangle.Next, the horizontal side between (3, 4) and (8, 4). Here, the y-coordinates are the same, so the distance is the difference in the x-coordinates. So, 8 - 3 equals 5 units. That's the other leg of the triangle.Now, the third side is between (3, -3) and (8, 4). This one isn't aligned with the axes, so I'll need to use the distance formula to find its length. The distance formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Plugging in the coordinates, we get sqrt[(8 - 3)^2 + (4 - (-3))^2] which simplifies to sqrt[5^2 + 7^2] = sqrt[25 + 49] = sqrt[74]. That's approximately 8.6 units.Now, let's check if it's a right triangle. The two legs are 5 and 7 units, and the hypotenuse is sqrt(74). According to the Pythagorean theorem, 5^2 + 7^2 should equal (sqrt(74))^2. Calculating, 25 + 49 equals 74, and (sqrt(74))^2 is also 74. So, yes, it satisfies the Pythagorean theorem, confirming that it's a right triangle.Since it's a right triangle, the area is simply half the product of the two legs. So, area = (1/2) * 5 * 7. Let's compute that: 5 times 7 is 35, and half of that is 17.5. So, the area of the triangle is 17.5 square units.Wait a minute, the problem asks to round the answer to the nearest tenth. But 17.5 is already to the nearest tenth, so I think that's the final answer. Just to make sure, let me try using the shoelace formula as another method to verify.The shoelace formula is given by:Area = (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|Plugging in the coordinates:x1 = 3, y1 = -3x2 = 8, y2 = 4x3 = 3, y3 = 4So, Area = (1/2) |3*(4 - 4) + 8*(4 - (-3)) + 3*(-3 - 4)|Simplify each term:3*(0) = 08*(7) = 563*(-7) = -21Adding them up: 0 + 56 - 21 = 35Taking absolute value: |35| = 35Then, half of that is 17.5So, both methods give me the same result, which is reassuring. Therefore, I'm confident that the area of the triangle is 17.5 square units.

answer：First, I need to calculate the value of ( M ) by dividing 1764 by 4.[M = frac{1764}{4} = 441]Next, I will determine ( N ) by dividing ( M ) by 4.[N = frac{441}{4} = 110.25]Finally, I will find ( X ) by subtracting ( N ) from ( M ).[X = 441 - 110.25 = 330.75]The value of ( X ) is 330.75.

answer：To find the center of the circle given by the equation ( x^2 + 4x + y^2 - 8y + 16 = 0 ), we follow these steps:1. **Rearrange the equation**: Group the ( x )-terms and ( y )-terms together. [ x^2 + 4x + y^2 - 8y = -16 ]2. **Complete the square** for both ( x ) and ( y ) terms: - For ( x ): ( x^2 + 4x ) becomes ( (x + 2)^2 - 4 ). - For ( y ): ( y^2 - 8y ) becomes ( (y - 4)^2 - 16 ).3. **Substitute back** into the equation: [ (x + 2)^2 - 4 + (y - 4)^2 - 16 = -16 ]4. **Simplify** the equation: [ (x + 2)^2 + (y - 4)^2 - 20 = -16 ] [ (x + 2)^2 + (y - 4)^2 = 4 ]5. **Identify the center** from the standard form of the circle equation ( (x - h)^2 + (y - k)^2 = r^2 ), where ( (h, k) ) is the center.The center of the circle is ( (-2, 4) ).