Question

The total number of relative maximum and minimum points of the function whose derivative is f ' (x) = x2(x + 1)3(x – 4)3 is (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 12. Find all absolute and relative

Answer 1

The function's derivative, f'(x) = x²(x + 1)³3(x - 4)³, has 2 relative maximum and minimum points.

So, the correct answer is C) 2.

Step-by-step explanation:

The function's derivative, f'(x) = x²(x + 1)³3(x - 4)³, gives us information about the critical points of the function. Relative maximum and minimum points occur where the derivative is zero or undefined. To find these points, we set the derivative equal to zero and solve for x: x²(x + 1)³3(x - 4)³ = 0. By analyzing the signs of the factors, we can determine the number of relative maximum and minimum points:

• When x = 0, both x² and (x + 1)³ are negative, while (x - 4)³ is positive. So, this point is a relative maximum.
• When x = -1, both x² and (x - 4)^3 are negative, while (x + 1)³ is positive. So, this point is a relative minimum.
• When x = 4, both (x + 1)³ and (x - 4)³ are positive, while x² is zero. As (x + 1)³and (x - 4)³ are both cubed, this point represents a saddle point rather than a relative maximum or minimum.

Therefore, the total number of relative maximum and minimum points is 2.

So, the correct answer is C) 2.

Answer 2

(D) 3

Step-by-step explanation:

A function f(x) has a relative minimum and maximum when its derivative f'(x) is equal to zero. Given f'(x) = x^2*(x + 1)^3*(x – 4)^3, f'(x) = 0 at x = 0, x = -1 and x = 4. Therefore, the total number of relative maximum and minimum points of f(x) is 3.