Find the absolute maximum and minimum of the function f(x)=sin^2x+cosx

1 answer:

6 0

Hello :
f(x) = sin²(x) +cos(x)
the dirivate is : f'(x) = 2sin(x) cos(x) -sin(x) because :
( sin²(x) )' = 2sin(x) cos(x) and (cos(x))' = - sin(x)

f'(x) = 0 : sin(x) (2cos(x) - 1) = 0
sin(x) = 0 : x = k π k in Z
2cos(x) - 1 =0 : 2cos(x) =1
cos(x) = 1/2 = cos( π /3)
x = π /3 +2k π or x = - π /3 +2k π
in intervall : [0 , 2π] :
k=0 : x= 0 f(0) = 1 ( minimum)
k=1 : x = π f(π) = -1 (minimum)
x= π /3 f ( π /3 ) = 1.25 ( maximum)

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