Question

Find the absolute maximum value of f(x) = cos x + sin² x, x ∈ [0, π].

Answer 1

f(x) = cos x + sin² x

f'(x) = –sinx + 2sinx·cosx

f'(x) = 0

–sinx + 2sinx·cosx = 0

sinx (2cosx – 1) = 0

sin x = 0 or 2cos x = +1

x = 0 or cos x = `1/2`

x = π or x = `pi/3`

f(x) = cos x + sin² x

f(0) = cos 0 + sin² 0

= 1 + 0 

= 1

`f(pi/3)=cospi/3+sin^2pi/3`

= `1/2+(sqrt3/2)^2`

= `1/2+3/4`

= `5/4`

f(π) = cosπ + sin²π

= –1 + (0)² 

= 1

∴ The absolute maximum value is `5/4`.