Answer
No such function exists
Work Step by Step
Suppose that such a function f exists.
The Mean Value Theorem states that if $f$ is a continuous function over the closed interval $[a,b]$ and differentiable over the open interval $(a,b)$, then there exists at least one point $c\in(a,b)$ so that
$$f′(c)=\dfrac{f(b)−f(a)}{b−a}.$$
By the Mean Value Theorem there is a number $c\in(0,2)$ with
$f'(c)$ = $\frac{f(2)-f(0)}{2-0}$ = $\frac{5}{2}$
But this is impossible since
$f'(x)\leq 2 \lt \frac{5}{2}$ for all $x$
so no such function exists.
