Q. 143.7( 3 Votes )
Find the absolute maximum and minimum values of the function f given by
f (x) = cos2 x + sin x, x ∈ [0, π]
Answer :
It is given that f (x) = cos2 x + sin x, x ∈ [0, π]
f’(x) = 2cosx( - sinx) + cosx
= - 2sinxcosx + cosx
Now, if f’(x) = 0
⇒ 2sinxcosx = cosx
⇒ cosx(2sinx - 1)=0
⇒ sin x = 
or cosx = 0
⇒ x = 
Now, evaluating the value of f at critical points x = 
and x =
and at the end points of the interval [0,π], (ie, at x = 0 and x =π), we get,
f
f(0)=
f(π)=cos2π + sinπ = ( - 1)2 + 0 =1
f
Therefore, the absolute maximum value of f is 
occurring at x = 
and the absolute minimum value of f is 1 occuring at x =1, 
and π.
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