What are all the critical points of f(x)=sinx+cosx ? 0=

The critical points are determined by differentiating the
function and equating the derivative to 0. It is solved to determine
x.

f(x) = sin x + cos x

f'(x)
= cos x - sin x = 0

=> cos x = sin
x

=> tan x =
1

=> x = arc tan
1

=> x = pi/4 ,
5*pi/4

At x = pi/4 , f(x) = sqrt
2

at x = 5*pi/4, f(x) = -sqrt
2

The critical points are at x = pi/4  and x
= 5*pi/4, and the extreme values are (pi/4, sqrt 2) and (5*pi/4,-sqrt
2).