Based on the information in the uploaded image, here is the analysis and solution:

1. Maximum Values of $f(x)$ and $g(x)$:

   • From the graph, the maximum value of $f(x)$ is $100$.
   • The maximum value of $g(x) = x^2(9 - x)$ is computed as follows:
     • $g(x) = x^2(9 - x)$ is a cubic function. To find its maximum, we differentiate and solve $g'(x) = 0$: $g'(x) = 2x(9 - x) - x^2 = 18x - 3x^2 = 3x(6 - x).$ The critical points are $x = 0$ and $x = 6$.
     • Evaluating $g(x)$ at $x = 6$: $g(6) = 6^2(9 - 6) = 36 \cdot 3 = 108.$ Thus, $g(x)$ has a maximum value of $108$, which is higher than the maximum value of $f(x)$.

2. Comparison of End Behavior:

   • For $f(x)$, as $x \to \infty$, $y \to -\infty$, as this is a downward-opening polynomial.
   • For $g(x) = x^2(9 - x)$, as $x \to \infty$, $y \to -\infty$, because the cubic term $-x^3$ dominates.

Correct Answer for End Behavior:

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

Let me know if you'd like a step-by-step explanation of any part!

Questions for Expansion:

1. Would you like a detailed derivation of the derivative for $g(x)$ to confirm its maximum point?
2. How do the graphs of $f(x)$ and $g(x)$ differ in behavior at $x \to 0^+$?
3. What is the role of the leading term in determining the end behavior of these polynomials?
4. Can we analyze the symmetry of $g(x)$ in comparison to $f(x)$?
5. Would you like a general discussion on finding maxima of polynomials?

Tip:

For any polynomial, the degree of the leading term is the key determinant of end behavior—always focus on this term for quick insights!