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

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)