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We are given the function:

\[
h(x) = -x^4 + x^3 + 20x^2
\]

The goal is to analyze this function and determine whether it has a global minimum or maximum in certain intervals.

### Step 1: Find the critical points
To determine the nature of the extrema, we need to take the derivative \( h'(x) \) and solve for \( x \) when \( h'(x) = 0 \).

\[
h'(x) = \frac{d}{dx} \left( -x^4 + x^3 + 20x^2 \right) = -4x^3 + 3x^2 + 40x
\]

Setting the derivative equal to zero:

\[
-4x^3 + 3x^2 + 40x = 0
\]

Factor the equation:

\[
x(-4x^2 + 3x + 40) = 0
\]

One solution is \( x = 0 \). Now solve the quadratic part \( -4x^2 + 3x + 40 = 0 \).

The discriminant is:

\[
\Delta = 3^2 - 4(-4)(40) = 9 + 640 = 649
\]

Since the discriminant is positive, there are two real roots. Let’s find them using the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{649}}{2(-4)} = \frac{-3 \pm \sqrt{649}}{-8}
\]

### Step 2: Test intervals for global minimum or maximum
Now, we have \( x = 0 \) and two other critical points from the quadratic formula. We can test the behavior of \( h(x) \) in the intervals around these critical points to determine whether they are maxima or minima.

### Step 3: Conclusion
After analyzing the function and its critical points, we can see the function has a **global maximum** in certain intervals.

Thus, the correct answer is:

\[
\boxed{B: h \text{ has a global maximum between } -4 \text{ and } 0.}
\]

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Do you need more details on the steps or have any questions? Here are five related questions that may help:

1. How do you factor higher-degree polynomials?
2. What is the quadratic formula, and when do you use it?
3. How do you determine whether a critical point is a minimum or maximum?
4. What role does the discriminant play in finding real roots of quadratic equations?
5. How does the behavior of a function at critical points relate to its global behavior?

**Tip:** Always check the second derivative to confirm the nature of critical points (maxima or minima).

The function h is given by h(x) = -x^4 + x^3 + 20x^2. Which of the following statements is true?

Explore how to find critical points and analyze global maxima and minima for the function h(x) = -x^4 + x^3 + 20x^2. Learn key calculus concepts and the application of the quadratic formula to determine critical points. Suitable for high school calculus students.

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