Let f,g:[a,b]⟶R continuous in [a,b] and differentiable in (a,b) such that f(a)=f(b)=0, then, theres exist c∈(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 φ to help me. I didn't want a solution of exercise, just a suggestion.
Answer
Use φ(x)=f(x)eg(x)
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