derivatives - Real Analysis. Suggestions.

Let $f,g: [a,b] \longrightarrow \mathbb{R}$ continuous in $[a,b]$ and differentiable in $(a,b)$ such that $f(a) = f(b) = 0$, then, theres exist $c \in (a,b)$ such that:

$$g'(c)f(c) + f'(c) = 0.$$

I tried to use the Rolle's Theorem and Means Value Theorem, but I couldn't define an auxiliary function $\varphi$ to help me. I didn't want a solution of exercise, just a suggestion.

Answer

Use

$$\varphi(x)=f(x)e^{g(x)}$$