Let f(x) = x^3 - 15x^2 - 33x - 36 be a real-valued function. Which of the following statements is/are TRUE? A. f(x) does not have a local maximum. B. f(x) has a local maximum. C. f(x) does not have a local minimum. D. f(x) has a local minimum.

GATE 2023 · Engineering Mathematics · Maxima Minima · medium

Answer: f(x) has a local maximum at x = -1 and a local minimum at x = 11. Statements B and D are TRUE.

1. Compute the first derivative: Differentiating f(x) = x^3 - 15x^2 - 33x - 36 term by term: f'(x) = 3x^2 - 30x - 33.
2. Find critical points: 3(x - 11)(x + 1) = 0 => x = -1 or x = 11.
3. Second derivative test: f''(x) = 6x - 30. At x = -1: f''(-1) = -6 - 30 = -36 < 0 => local maximum. At x = 11: f''(11) = 66 - 30 = 36 > 0 => local minimum.