1.Let $a, b$ be two positive numbers, and let $f:[a, b] \rightarrow \mathbf{R}$ be a continuous function, differentiable on $(a, b)$. Prove that there exists $c \in(a, b)$ such that$$\frac{1}{a-b}(a f(b)-b f(a))=f(c)-c f^{\prime}(c)$$2. Let $f:[a, b] \rightarrow \mathbf{R}$ a continuous positive function, differentiable on $(a, b)$. Prove that there exists $c \in(a, b)$ such that$$\frac{f(b)}{f(a)}=e^{(b-a) \frac{f^{\prime}(c)}{f(c)}}$$3.Let $f, g:[a, b] \rightarrow \mathbf{R}$ be two continuous functions, differentiable on $(a, b)$. Assume in addition that $g$ and $g^{\prime}$ are nowhere zero on $(a, b)$ and that $f(a) /$$g(a)=f(b) / g(b)$. Prove that there exists $c \in(a, b)$ such that$$\frac{f(c)}{g(c)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}$$4. Let $f:[a, b] \rightarrow \mathbf{R}$ be a continuous function, differentiable on $(a, b)$ and nowhere zero on $(a, b)$. Prove that there exists $\theta \in(a, b)$ such that$$\frac{f^{\prime}(\theta)}{f(\theta)}=\frac{1}{a-\theta}+\frac{1}{b-\theta}$$
The "type" of problems above all follow the same kind of approach, that is first finding a suitable function and then applying Mean Value Theorem on that function, so it often just boils down to finding the suitable function. Some times it's easy but other times it may not be easy, so what are some strategy/technique to find that suitable function?
Like in this post, which is a similar kind of problem as the above problems, but the solution in the thread constructed a suitable function, whose motivation seems unclear, so how do people find such suitable functions?