https://file.aiquickdraw.com/mathbot/file/172684860403039tzr7yx.jpg

The function provided is:

\[
h(x) = 2x^5 - 5x^4 - x^3 + 13x - 7
\]

We need to determine the behavior of the graph to assess whether it has a global maximum, global minimum, both, or neither.

### Step-by-Step Analysis:

1. **Degree of the Polynomial:**
   The degree of the polynomial is 5, which is odd. For odd-degree polynomials, the end behavior is different on each side:
   - As \( x \to \infty \), \( h(x) \to \infty \) (because the leading coefficient \( 2 \) is positive).
   - As \( x \to -\infty \), \( h(x) \to -\infty \).

2. **Behavior at Extrema:**
   A degree-5 polynomial can have up to 4 turning points, which represent local maxima or minima. However, due to the end behavior, the function does **not** have a global maximum (since it increases without bound as \( x \to \infty \)).

3. **Global Minimum:**
   Since \( h(x) \to -\infty \) as \( x \to -\infty \), it’s possible for the function to attain a global minimum at a point where it reaches the lowest value before increasing again.

Given this information, the function **has a global minimum but not a global maximum**.

### Conclusion:
The correct answer is:
- **The function \( h \) has a global minimum but not a global maximum.**

Would you like any further details or clarification on this?

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Here are five related questions to expand your understanding:
1. How do you determine local maxima and minima for a polynomial function using derivatives?
2. What is the significance of the degree of a polynomial on its end behavior?
3. How can we find the critical points of a polynomial function like \( h(x) \)?
4. How would the behavior of the graph change if the leading coefficient were negative?
5. How can you graph a polynomial of degree 5 to verify the presence of global extrema?

**Tip:** Always check the end behavior of polynomial functions to understand if they have global maxima or minima.

The polynomial h is given by h(x) = 2x^5 - 5x^4 - x^3 + 13x - 7. Which of the following statements fits the description of the graph of h?

Analyze the polynomial function h(x) = 2x^5 - 5x^4 - x^3 + 13x - 7 and determine whether it has a global maximum, minimum, or both. This problem covers polynomial behavior, extrema, and the relationship between degree and end behavior.

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