Let's go through each part of the problem step by step.

Given function:

$f(x) = x(24 - x)$

(A) Find $f(x + h)$

We substitute $x + h$ for $x$ in the function $f(x)$: $f(x + h) = (x + h)(24 - (x + h))$

Now expand and simplify the expression: $f(x + h) = (x + h)(24 - x - h) = x(24 - x - h) + h(24 - x - h)$ $f(x + h) = x(24 - x - h) + h(24 - x - h) = x(24 - x) - xh + h(24 - x) - h^2$ $f(x + h) = (24x - x^2 - xh) + (24h - hx - h^2)$ $f(x + h) = 24x - x^2 - xh + 24h - hx - h^2$ Since $-xh$ and $-hx$ are like terms, we combine them: $f(x + h) = 24x - x^2 - 2xh + 24h - h^2$

So: $f(x + h) = 24x - x^2 - 2xh + 24h - h^2$

(B) Find $f(x + h) - f(x)$

Subtract $f(x)$ from $f(x + h)$: $f(x + h) - f(x) = (24x - x^2 - 2xh + 24h - h^2) - (24x - x^2)$

Simplify by canceling out like terms: $f(x + h) - f(x) = -2xh + 24h - h^2$

(C) Find $\frac{f(x + h) - f(x)}{h}$

Divide the expression from part (B) by $h$: $\frac{f(x + h) - f(x)}{h} = \frac{-2xh + 24h - h^2}{h}$

Simplify by dividing each term by $h$: $\frac{f(x + h) - f(x)}{h} = -2x + 24 - h$

Thus, the final answer is: $\frac{f(x + h) - f(x)}{h} = -2x + 24 - h$

Would you like more details or have any questions?

Here are five related questions to further your understanding:

1. What happens to the expression $\frac{f(x + h) - f(x)}{h}$ as $h$ approaches 0?
2. How does the expression $\frac{f(x + h) - f(x)}{h}$ relate to the derivative of $f(x)$?
3. Can you find the derivative of $f(x) = x(24 - x)$ using the power rule?
4. What is the significance of the term $h^2$ in the expansion?
5. How would the problem change if $f(x)$ were a different quadratic function?

Tip: The expression $\frac{f(x + h) - f(x)}{h}$ is the difference quotient, a fundamental concept in calculus used to define the derivative of a function.