Let the function f(theta) = | sin(theta) cos(theta) tan(theta) | | sin(theta/3) cos(theta/3) tan(theta/3) | where f'(theta) denote the derivative of f with respect to theta. Which of the following statements is/are TRUE? I. There exists theta in (5*pi/8, 7*pi/8) such that f'(theta) = 0 II. There exists theta in (11*pi/16, 13*pi/16) such that f'(theta) = 0 A. I only B. II only C. Both I and II D. Neither I nor II

GATE 2014 · Engineering Mathematics · Differentiation · medium

Answer: Both I and II are TRUE. Answer: C.

  1. Simplify f(theta): The 2x2 determinant from the first two columns gives sin(theta)*cos(theta/3) - cos(theta)*sin(theta/3) = sin(theta - theta/3) = sin(2*theta/3). So f(theta) = sin(2*theta/3).
  2. Find f'(theta): f'(theta) = (2/3)*cos(2*theta/3). Setting f'(theta) = 0: cos(2*theta/3) = 0, so 2*theta/3 = pi/2, giving theta = 3*pi/4.
  3. Check Statement I: theta = 3*pi/4 in (5*pi/8, 7*pi/8)?: 3*pi/4 = 6*pi/8, and 5*pi/8 < 6*pi/8 < 7*pi/8. YES, Statement I is TRUE.
  4. Check Statement II: theta = 3*pi/4 in (11*pi/16, 13*pi/16)?: 3*pi/4 = 12*pi/16, and 11*pi/16 < 12*pi/16 < 13*pi/16. YES, Statement II is TRUE.