Question:

Step 1: Determine the set A by solving the inequality. We need to find all values of $x$ satisfying $x^2 + 20 \leq 9x$. Rearranging the inequality: $$x^2 - 9x + 20 \leq 0$$ Factoring the quadratic expression: $$(x-4)(x-5) \leq 0$$ For a product of two factors to be non-positive, the factors must have opposite signs (or one equals zero). This occurs when $x$ is between the roots: $$A = [4, 5]$$ Step 2: Find the critical points of $f(x)$ by computing its derivative. We have $f(x) = 2x^3 - 15x^2 + 36x - 48$. Taking the derivative: $$f'(x) = 6x^2 - 30x + 36$$ Factoring out the common factor: $$f'(x) = 6(x^2 - 5x + 6) = 6(x-2)(x-3)$$ Step 3: Analyze the sign of $f'(x)$ on the interval $[4, 5]$. The critical points are $x = 2$ and $x = 3$, both of which lie outside our domain $[4, 5]$. For any $x \in [4, 5]$: - $(x-2) > 0$ (since $x \geq 4 > 2$) - $(x-3) > 0$ (since $x \geq 4 > 3$) Therefore, $f'(x) = 6(x-2)(x-3) > 0$ for all $x \in [4, 5]$. This means $f(x)$ is strictly increasing on the interval $[4, 5]$. Step 4: Find the maximum value on $[4, 5]$. Since $f(x)$ is increasing on $[4, 5]$, the maximum occurs at the right endpoint $x = 5$. Computing $f(5)$: $$f(5) = 2(5)^3 - 15(5)^2 + 36(5) - 48$$ $$f(5) = 2(125) - 15(25) + 180 - 48$$ $$f(5) = 250 - 375 + 180 - 48$$ $$f(5) = 7$$ The maximum value of $f(x)$ on set $A$ is $\boxed{7}$, which corresponds to **Option 2**.