Question

For what values of x is `f(x)= x-cosx` concave or convex?

Answer 1

`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.

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.