f(x) is continous in [0,1], differentiable in (0,1), and f(0)=0,f(1)=1.

prove that for any positive numbers a and b, there exist < in (0,1) such that

.

how to prove this kind of problem about " there exist ..."?

I know there is three theorems in the textbook may be helpful.

But here there are two parameters.

Any suggestion, hint or solution is greatly appreciated!