Analysis of the Problem:

We are tasked with comparing the maximum values of $f(x)$ and $g(x)$, as well as comparing their end behavior when $x > 0$.

From the given information:

• $g(x) = x^2 (9 - x)$
• The graph shows $f(x)$, a polynomial function, with its maximum value and shape provided.

Step 1: Compare Maximum Values

$f(x)$

• From the graph of $f(x)$, the maximum value occurs at $y = 100$ when $x > 0$.

$g(x)$

1. The function $g(x) = x^2 (9 - x)$ is a cubic polynomial.
2. To find the maximum value of $g(x)$ for $x > 0$, take the derivative: $g'(x) = 2x(9 - x) - x^2 = 18x - 3x^2$
3. Set $g'(x) = 0$: $18x - 3x^2 = 0 \implies x(18 - 3x) = 0$ Solving gives $x = 0$ or $x = 6$.
4. Evaluate $g(x)$ at $x = 6$: $g(6) = 6^2 (9 - 6) = 36 \cdot 3 = 108$

Thus, the maximum value of $g(x)$ is 108, which is greater than the maximum value of $f(x) = 100$.

Step 2: Compare End Behavior

End Behavior of $f(x)$

• From the graph, as $x \to \infty$, $f(x) \to -\infty$.

End Behavior of $g(x)$

• Analyze the degree and leading term of $g(x) = x^2 (9 - x)$: $g(x) = -x^3 + 9x^2$ The leading term is $-x^3$, which dominates as $x \to \infty$. Thus: $\text{As } x \to \infty, g(x) \to -\infty.$

Conclusion on End Behavior:

• For both $f(x)$ and $g(x)$, as $x \to \infty$, $y \to -\infty$.

Final Answers:

1. Maximum values:

   • The maximum value of $f(x)$ is 100, which is less than the maximum value of $g(x) = 108$.

2. End behavior:

   • The correct choice is C: "For both functions, when $x \to \infty, y \to -\infty$."

Do you want details or have further questions?

Here are 5 related questions for deeper exploration:

1. How do we confirm that $g(x) = x^2 (9 - x)$ has no other maximum points for $x > 0$?
2. Can we determine the specific point at which $f(x)$ achieves its maximum value?
3. How does the degree of a polynomial affect its end behavior?
4. What are the intercepts of $g(x)$, and how do they compare to those of $f(x)$?
5. How do the critical points of $g(x)$ influence its shape and maximum value?

Tip: Always double-check derivatives and critical points when analyzing maximum or minimum values for functions!