Rolle-like problem

Question

Let $f:[a,b]\to \mathbb{R}$ a function that is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in (a,b)$ such that $$f'(c)\cdot \int\limits_a^c f(x) ~dx=-f^2(c).$$ This kind of problems usually reduces to find an appropriate function and to apply Rolle theorem for this function, in order to get the required equality. The obvious/easy choices did not get me there.

Answer 1

This is not true, as the example $a=0,b=1$ and $f(x)=x$ shows.

But if we have in addition that $f(b)=0$, then define

$g(x):= f(x)\int\limits_a^x f(t) ~dt$. Then $g(a)=g(b)=0$ and

$$g'(x)=f'(x)\int\limits_a^x f(t) ~dt+f(x)^2.$$

Answer 2

You claim seems incorrect. take $$f(x)=1$$ then $$f(c)=0$$

Answer 3

It might be easier to take $F'(x)=f(x)$ so: $$F''(c)\left[F(c)-F(a)\right]=-\left[F'(c)\right]^2$$ now can such a point exist?