Question

Find the absolute maximum and minimum values of the function f given by f(x) = cos²x + sin x, x ∈ [0, π].

Answer 1

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 = \(\frac{\pi}{6}\) or \(\frac{\pi}{2}\) as x ∈ [0,π] 

So, 

The critical points are x = \(\frac{\pi}{2}\) and x = \(\frac{\pi}{6}\) and at the end point of the interval [0,π] we have,

Thus, 

We conclude that the absolute maximum value of f is 5/4 at x = \(\frac{\pi}{6}\) and absolute minimum value of f is 1 which occurs at x = 0,\(\frac{\pi}{2}\) and π