Question

The function h is given by h(x)=-x⁴+x³+20x². Which of the following statements is true?

A. h has a global minimum between x=-4 and x=0.
B. h has a global maximum between x=-4 and x=0.
C. h has a global minimum between x=0 and x=5.
D. h has a global maximum between x=0 and x=5.

Answer 1

The function h(x) = -x⁴+x³+20x² has a global maximum between x = -4 and x = 0 at x = -5.

Step-by-step explanation:

The function h(x) = -x⁴+x³+20x² is a polynomial function. To determine whether it has a global minimum or maximum between two given values of x, we need to analyze its critical points and the concavity of the function.

1. First, take the first derivative of h(x) to find the critical points:

• h'(x) = -4x³ + 3x² + 40x.

Setting h'(x) = 0, we can solve for the critical points:

• -4x³ + 3x² + 40x = 0.

Using the quadratic formula, we can find the solutions:

• x = 0, -5, and 4.

To determine whether these critical points are local minimums or maximums, we need to check the second derivative of h(x).

• h''(x) = -12x² + 6x + 40.

Checking the second derivative at each critical point:

• h''(0) = 40, so there is no local minimum or maximum at x = 0.
• h''(-5) = -110, so there is a local maximum at x = -5.
• h''(4) = 168, so there is a local minimum at x = 4.

Therefore, between x = -4 and x = 0, h has a global maximum at x = -5.