Reversing sign of third derivative

Question

I don't understand some part of the solution given to this question:

I can understand how the sign of $f''(x)$ can be reversed (i.e. flipping over the x-axis) but I don't get why changing $x$ to $-x$ would change the sign of $f'''(x)$?

Answer 1

Let $g(x)=f(-x)$. Then using the chain rule in each step, we get $$g'(x)=-f'(-x),$$ $$g''(x)=-(-f''(-x))=f''(-x),$$ and finally $$g'''(x)=-f'''(-x).$$ So if $f'''$ is always positive then $g'''$ is always negative, and vice versa.