GATE CS 2021 Set 2 Q.35 (CS Q.25): Calculus

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a continuous function on the interval $[-3, 3]$ and a differentiable function in the interval $(-3, 3)$ such that for every $x$ in the interval, $f'(x) \le 2$ . If $f(-3) = 7$ , then $f(3)$ is at most ______.

Your answer

Official answer and solution

Official answer: 19
From: Official GATE 2021 answer key

Solution

Apply the Mean Value Theorem on the continuous and differentiable function $f(x)$ over the interval $[-3, 3]$ .

Since $f$ is continuous on $[-3, 3]$ and differentiable on $(-3, 3)$ , by the Mean Value Theorem there exists some $c \in (-3, 3)$ such that:

\frac{f(3) - f(-3)}{3 - (-3)} = f'(c)

We are given that $f'(x) \le 2$ for all $x \in (-3, 3)$ . Therefore:

\frac{f(3) - f(-3)}{6} \le 2

f(3) - f(-3) \le 12

Substituting $f(-3) = 7$ :

f(3) - 7 \le 12 \implies f(3) \le 19

Thus, $f(3)$ is at most 19.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 35 of 65, printed as CS Q.25
Section: Computer Science and Information Technology

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