Find the absolute maximum and minimum values of the function f given by f(x) = cos 2 x + sin x, x ∈ [0, π].

Question

Philoid · Accepted Answer

f(x) = cos²x + sin x

f’(x) = 2 cos x (–sin x) + cos x

= 2 sin x cos x + cos x

now, f’(x) = 0

⇒ 2 sin x cos x = cos x

⇒ cos x(2sin x – 1) = 0

⇒ sin x = 1/2 or cos x = 0

⇒ x = or as x [0, ]

So, the critical points are x = and x = and at the end point of the interval [0, ] we have,

f() = = 5/4

f(0) = = 1 + 0 = 1

f(π) = = (–1²) + 0 = 1

f() = = 0 + 1 = 1

Thus, we conclude that the absolute maximum value of f is 5/4 at x = , and absolute minimum value of f is 1 which occurs at x = 0, .