Question

Let $f(x)$ be a continuous and differentiable function on $[3,18]$. If $f(3) = -50$ and $f'(x) \le 20$, then the largest possible value of $f(18)$ is __________ (in integer).

Hint:

When $f'(x)$ has an upper bound, the function can grow at most at that rate. Use MVT.

Answer 1

Correct Answer: 250

From the Mean Value Theorem:
\[ f(18) - f(3) \le 20 \times (18 - 3). \]
Compute:
\[ f(18) - (-50) \le 20 \times 15 = 300. \]
Thus:
\[ f(18) \le 300 - 50 = 250. \]
Therefore, the maximum possible value is:
\[ \boxed{250} \]