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I am struggling with this problem. I don't want the solution only ideas on how to approach it. I'll tell you what I have tried so far. (High school knowledge).
The problem:
Consider a function g: which is 2 times differentiative and for which the following are true:
by Bolzano at [g(0),g(1)] [assuming g(0)<g(1) but we can do the same if g(1)<g(0) ]
the inner function is continuous therefore the integral is a differintiative function and also continuous.
s(g(0))*s(g(1)) < 0 (from (2))
so there there is a number k in (g(0),g(1)) such that s(k) = 0 although I don't know how this helps.
Another try in the same problem was to use Rolle theorem on [0,1] in one original function of h which is:
+
but again I couldn't prove that f(0) = f(1) to apply the theorem.
Any ideas?
(1)
(2)

