Not a standard inequality

Consider $h(x) = e^{g(x)} f(x)$ , then $h'(x) = e^{g(x)}(f'(x) + f(x)g'(x)) \geq 0$ for all real $x$ , so $h(x)$ is non-decreasing.

Thus $\frac{f(0)}{f(1)} = \frac{h(0)}{h(1)} e^{g(1)-g(0)} \leq e^{g(1)-g(0)}$ since $h$ is non-decreasing.

But $g(1)-g(0) = \int_{0}^{1} g'(x) dx \leq \int_{0}^{1} x dx = \frac{1}{2}$ since $x \geq g'(x)$ .

Hence $M = e^{0.5}$ , and the answer is $1000M^{2} = 2718.28...$

The equality case is left to the reader. (Hint: we need $g'(x) = x$ and $h(x)$ is constant)