y = 3/4 cos (2x − π/3) for x ∈ [− π/2 , π].

Question

Determine all values of x within the given domain where the function reaches its minimum
and maximum values.

Answer 1

y=¾cos(2x-π/3), dy/dx=-(3/2)sin(2x-π/3)=0 at turning points.

Therefore, 2x-π/3=πn where n is an integer, 2x=πn+π/3, x=πn/2+π/6.

n=-1: x=-π/2+π/6=-⅓π, y=¾cos(-π)=¾cos(π)=-¾

n=0: x=⅙π, y=¾cos(0)=¾

n=1: x=π/2+π/6=⅔π, y=¾cos(π)=-¾

So we have three turning points: (-⅓π,-¾), (⅙π,¾), (⅔π,-¾).

Since y goes from - to + to - between these points, these are:

MIN(-⅓π,-¾), MAX(⅙π,¾), MIN(⅔π,-¾).