Constructing a new function $F(x)$ and apply Rolle's Theorem

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I have tried to construct a new function $F(x)$ such that $F(a)=F(b)=0$ and apply Rolle's Theorem, but the question is: there's two variables $f(c)$ and $g(c)$ on the rate of change (left-hand side) that I am concerned at. So how do I construct it?

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This is my attempt and it's correct I think.

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