Question

How do you find the critical points of `f(x)=sinx+cosx`?

Answer 1

`y = sin x + cos x`
Use the Trig Identity `sin + cos x = sqrt{2} sin (x + pi/4)`.
`y = sqrt{2} sin (x + pi/4)`
`y` min when `sin (x + pi/4) = -1 rArr x + pi/4 = 3/2 pi rArr x = 5/4 pi`.
`y` max when `sin(x + pi/4) = 1 rArr x + pi/4 = sin pi/2 rArr x = pi/4`.
In the interval `(0, 2 pi)` there are `2` answers: `pi/4` and `5/4 pi`.
Check
When `x = pi/4 rArr y = sqrt{2}/2 + sqrt{2}/2 = sqrt{2}` (Max)
When `x = 5/4 pi rArr y = -sqrt{2}/2 - sqrt{2}/2 = - sqrt{2}` (Min)