Convex downward function and its inverse function How to prove that if function $f$ is convex downward and invertible then $f^{-1}$ is convex downward or convex upward? When is it downward and when upward?

Let $g$ be the inverse function of $f$. That is , $g=f^{-1}$.

Then, $$f(g(x))=x$$ Differentiating, $$f'(g(x)).g'(x)=1$$ $$g'(x)=\frac{1}{f'(g(x))}$$ Differentiating again, $$g''(x)=\frac{-1}{\left(f'(g(x))\right)^2}.f''(g(x)).g'(x)$$

Substituting the value of $g'(x)$ obtained earlier, $$g''(x)=\frac{-f''(g(x))}{(f'(g(x)))^3}$$

You can draw your conclusions from here now.