Mean value theorem question. Show that if $f(2) = -2$ and $f ' (x) \geq 5$ for $x > 2$, then $f(4) \geq 8$.
I know that the $b$ value is $4$ and the $a$ value is $2$ for the mean value theorem equation. 
$$f'(c)=\frac{f(4) - f(2)}{4-2}$$
We know the value of $f(2)$ but for $f(4)$ I'm confused there and with the $f ' (x) \geq 5$ for $x > 2$.
Any help will be appreciated. 

Your equation can be rewritten as
$$f(4)=f(2)+(4-2)f'(c).\tag{1}$$
We also know that $c$ is between $2$ and $4$, and therefore in particular $c\gt 2$. 
So $f'(c)\ge 5$. Also, $f(2)=-2$. It follows from (1) that
$$f(4)\ge -2+(4-2)(5).$$
