Solutions

Suppose that $f'$ is increasing. Pick any point $(x_0,f(x_0))$. Notice that the tangent line of $f$ at this point is

$$g(x)=f'(x_0)(x-x_0)+f(x_0).$$

Let $h(x)=f(x)-g(x)$. For contradiction, suppose the tangent line intersects $f$ at another point $(a,f(a))$ with $a>x_0$ (the same contradiction would arise if $a<x_0$). Then $h(x_0)=f(x_0)-g(x_0)=0$ and $h(a)=f(a)-g(a)=0$ so by Rolle's theorem,

$$0=h'(y)=f'(y)-g'(y)=f'(y)-f'(x_0)$$

for some $y\in(x_0,a)$ but this contradicts the premise that $f'$ is increasing.