Find smallest possible value for f(4) given f(1) and f'(x) > 2

Suppose $f(1)=10$ and $f^{\prime }(x)\ge 2$ for all $x\in R$. Determine the smallest possible value for $f(4)$.

Proof

1. Note that $f$ is differentiable as $f^{\prime }(x)\ge 2$
2. Note that $f$ is continuous as $f$ is differentiable
3. Then by MVT, there exists a $c\in (1,4)$ such that $f^{\prime }(c)=\frac{f(4)-f(1)}{4-1}$
4. $\frac{1}{3}(f(4)-10)=f^{\prime }(c)\ge 2$ as $f^{\prime }(x)\ge 2$ for all $x\in R$
5. $f(4)\ge 16$