Verify MVT hypotheses and find the guaranteed value of c

First compute the slope of the secant line: $\frac{f(2)-f(0)}{2-0} = \frac{e^2 - e^0}{2} = \frac{e^2 - 1}{2}$. Since $f(x)=e^x$ is continuous everywhere and differentiable everywhere, both hypotheses of the MVT are satisfied on $[0,2]$, so a valid $c$ in $(0,2)$ must exist.

Since $f'(x) = e^x$, we need $e^c = \frac{e^2-1}{2}$. Taking the natural logarithm of both sides gives $c = \ln\left(\frac{e^2-1}{2}\right)$. Numerically this is approximately $c \approx 1.13$, which indeed lies in $(0,2)$ as required.