Let $f(x) = x^5 + x^3 +2x +1$ and let $g$ be its inverse. Let $a,b \in\mathbb R$ with $a < b$. Show that $g(b) − g(a) \le 0.5(b −a)$

Question

Let $f(x) = x^5 + x^3 +2x +1$ and let $g$ be its inverse. Let $a,b \in\mathbb R$ with $a < b$. Show that $g(b) − g(a) \le 0.5(b −a)$

I know that $f(x)$ is both continuous and differentiable, so it's possible to use the mean value theorem on it. The inequality also suggests the use of the mean value theorem on the inverse function, but I don't know how to proceed.

Any help would be kindly appreciated.

Answer 1

$f'(x)=5x^{4}+3x^{2}+2 \geq 2$. This implies $g'(x)=\frac 1 {f'(g(x))}\leq \frac 1 2$. Now apply MVT.