The existence of the interior point x0 in 1. makes f'(x0) = 1.

2. According to Buda's theorem, f'([0, 1]) contains [x 1, x2], where 0.

3. Structure

h(t)= a/(x 1+t( 1-x 1))+b/( 1+t(x2- 1))，0 & lt= t & lt= 1

Then h (0) = a/x1+b >; a+b

h( 1)= a+b/x2 & lt； a+b

So there is t0 to make h (t0) = a+b.

Note that the denominator of h(t) is the derivative somewhere, and the conclusion holds.