Is f(x)=xcosx concave or convex at x=pi/2?

1 Answer

function is concave at x=\pi/2

Explanation:

Given function:

f(x)=x\cos x

f'(x)=x(-\sin x)+\cosx

=-x\sin x+\cos x

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

f''(x)=-x\cos x-2\sin x

f''(pi/2)=-\pi/2\cos (\pi/2)-2\sin (\pi/2)

=0-2(1)

=-2

Since f(\pi/2)<0 hence the given function is concave at x=\pi/2