For what values of x is f(x)= x-cosxf(x)=xcosx concave or convex?

1 Answer
Jan 16, 2016

f(x)f(x) is convex on ((-pi)/2+2kpi,pi/2+2kpi)(π2+2kπ,π2+2kπ) and concave on (pi/2+2kpi,(3pi)/2+2kpi)(π2+2kπ,3π2+2kπ) where kk is an integer.

Explanation:

Concavity is determined by the sign of the second derivative:

  • If f''(a)>0, then f(x) is convex at x=a.
  • If f''(a)<0, then f(x) is concave at x=a.

First, determine the second derivative.

f(x)=x-cosx
f'(x)=1+sinx
f''(x)=cosx

So, we need to determine when cosx is positive and when it is negative. The sign of cosx will change whenever cosx=0.

This occurs when x=(-pi)/2,pi/2,(3pi)/2,(5pi)/2, and so on, increasing in intervals of pi.

cosx>0 on ((-pi)/2+2kpi,pi/2+2kpi) where k is an integer.
cosx<0 on (pi/2+2kpi,(3pi)/2+2kpi) where k is an integer.

Thus,

f(x) is convex on ((-pi)/2+2kpi,pi/2+2kpi) and concave on (pi/2+2kpi,(3pi)/2+2kpi) where k is an integer.