How do you find the intervals of increasing and decreasing using the first derivative given y=sinx+cosxy=sinx+cosx?

1 Answer
Jul 25, 2017

The function is increasing on intervals of the form [-(3pi)/4+2npi,pi/4+2npi][3π4+2nπ,π4+2nπ] and decreasing on intervals of the form [pi/4+2npi,(5pi)/4+2npi][π4+2nπ,5π4+2nπ], where n=0,\pm 1,\pm 2,...

Explanation:

The derivative of y=sin x+cos x is dy/dx=cos x-sin x. Setting this equal to zero yields cos x=sin x (which is equivalent to tan x=1). This occurs when x=...,-(7pi)/4,-(3pi)/4,pi/4,(5pi)/4,....

You can check that cos x>sin x (so dy/dx>0) which occurs when x\in (-(3pi)/4,pi/4), when x\in ((5pi)/4,(9pi)/4), etc...).

You can check that cos x < sin x (so dy/dx<0) which occurs when x\in ((-7pi)/4,-(3pi)/4), when x\in (pi/4,(5pi)/4), etc...).

The answer above follows from these observations.

You can confirm this visually by looking at the graph.

graph{sin(x)+cos(x) [-10, 10, -5, 5]}