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