Correct Answer - Option 3 : Both local minimum and local maximum
Concept:
For any function f(x), local minima occur at x = a, if f'(a) = 0 and f''(a) > 0, and local maxima occur at x = a, if f'(a) = 0 and f''(a) < 0.
Calculation:
A function f : (0, π) → R defined by f(x) = 2 sin x + cos 2x
Differentiate f(x) with respect to x
f'(x) = 2 cos x - 2sin 2x
For local minima and maxima f'(x) = 0
2 cos x - 2sin 2x = 0
cos x - sin 2x = 0
cos x - 2sin x cos x = 0
cos x (1 - 2sin x) = 0
cos x = 0 or 1 - 2sin x = 0
cos x = 0 or sin x = 1/2
\(x = \frac{\pi }{2}\) or \(x = \frac{\pi }{6}or\frac{{5\pi }}{6}\)
f'(x) = 2 cos x - 2sin 2x differentiate it again
f''(x) = -2sin x - 4cos 2x
At \(x = \frac{\pi }{2}\) ,
f''(π/2) = -2sin (π/2) - 4cos (2π/2) = -2×1 - 4×-1 = -2 + 4 = 2 > 0
so at x = π/2 local minima occur.
At x = π/6,
f''(π/6) = -2sin (π/6) - 4cos (2π/6) = -2×1/2 - 4×1/2 = -1 - 2 = -3 < 0
so at x = π/6 , local maxima occur.
At x = 5π/6,
f''(5π/6) = -2sin (5π/6) - 4cos (2×5π/6) = -2×1/2 - 4×-1/2 = -1 + 2 = 1> 0
so at x = 5π/6 local minima occur.
