a. Find the points of local maxima and minima, if any, of the following function defined in 0 <= x <= 6: f(x) = x^3 - 6x^2 + 9x + 15 b. Integrate: int_{0}^{pi/2} cos(x) / (1 + sin(x)) dx

GATE 1998 · Engineering Mathematics · Integration · medium

Answer: Part a: local max at x=1 (f=19), local min at x=3 (f=15). Part b: ln 2

1. Find critical points of f(x): f'(x)=0 gives x=1 and x=3. Both are in [0,6].
2. Classify critical points: f''(1) = 6 - 12 = -6 < 0 => local maximum at x=1, f(1) = 1-6+9+15 = 19. f''(3) = 18-12 = 6 > 0 => local minimum at x=3, f(3) = 27-54+27+15 = 15.
3. Evaluate the definite integral: int_{0}^{pi/2} cos(x)/(1+sin(x)) dx = int_{1}^{2} du/u = [ln u]_{1}^{2} = ln 2 - ln 1 = ln 2.