Suppose f,g, & h are defined on (a,b) & a<x0<b. Assume f and h are differentiable at x0, f(x0)=h(x0), & f(x)<g(x)<h(x) for all x in (a,b). Prove that g is differentiable at x0 and f'(x0)=g'(x0)=g'(x0).
It is important that $\displaystyle f(x_0)=h(x_0)$ so $\displaystyle -f(x_0)=-h(x_0)$.
If $\displaystyle \delta>0$ can you show that
$\displaystyle \frac{f(x_0+\delta)-f(x_0)}{\delta}<\frac{g(x_0+\delta)-g(x_0)}{\delta}<\frac{h(x_0+\delta)-h(x_0)}{\delta}~?$
There is a similar statement if $\displaystyle \delta<0$.
Those are used on the limit definition of the derivatives and $\displaystyle f'(x_0)=h'(x_0)$.