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 $\displaystyle x_1$ < $\displaystyle x_2$ in (0,1) such that

$\displaystyle \frac{a}{f'(x_1)}+\frac{b}{f'(x_2)}=a+b$.

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!