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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.

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

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  • $\begingroup$ How can one find such function? $\endgroup$ – Ooker Feb 25 '18 at 6:36
  • $\begingroup$ @Ooker: By observing that the desired equation looks like a first-order ODE. Netchaiev's solution looks like the anti-derivative used in the solution of the first-order ODE. $\endgroup$ – user21820 Feb 25 '18 at 7:52

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