Suppose that f : R -> 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 (-3, 3), |f'(x)| <= 2. If f(-3) = 7, then f(3) is at most __________.

GATE 2021 · Engineering Mathematics · Continuity · medium

Answer: 19

1. Apply the Mean Value Theorem: By MVT, since f is continuous on [-3, 3] and differentiable on (-3, 3), there exists c in (-3, 3) with f(3) - f(-3) = f'(c) * 6.
2. Apply the derivative bound: Since |f'(c)| <= 2, we get |f(3) - f(-3)| <= 12, which means f(3) - 7 <= 12, so f(3) <= 19.