Let’s analyze the problem based on the uploaded images.

(b) At which $x$-values does the function have local minima?

From the graph, local minima are the $x$-values where the curve reaches a low point before increasing again. These occur at approximately:

• $x \approx -8$
• $x \approx 3$

(c) What is the sign of the function's leading coefficient?

The leading coefficient of a polynomial determines the end behavior of the graph:

• As $x \to \pm\infty$, the graph of this polynomial approaches $-\infty$ on both ends, which indicates a negative leading coefficient.

(d) Which of the following is a possibility for the degree of the function?

The number of turning points on the graph is one less than or equal to the degree of the polynomial. The graph has 6 turning points, so the degree is at least $7$. Possible degrees are odd numbers greater than or equal to $7$:

• Degree possibilities: $7, 9$.

Summary of Answers:

• (b): $x \approx -8, 3$
• (c): Negative
• (d): $7, 9$

Would you like detailed explanations for any part?

Relative Questions:

1. How do turning points relate to the degree of a polynomial?
2. Why does the leading coefficient determine end behavior?
3. Can there be a degree less than 7 for this polynomial? Why not?
4. How can you verify local minima using calculus?
5. What happens to the graph of a polynomial as the degree increases?

Tip:

Always check the end behavior of a graph to deduce the sign of the leading coefficient!