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:
  • image (1)
  • imageimage (2)
a)Prove that g'(x) is strictly increasing. b)Prove that there is at least one number p such that and also that What I've tried so far: for a) Suppose such that then from (1) => ... 0< 0 Reductio ad absurdum I tried to arrive at contradiction assuming g''(x) < 0 because then the only possibility would be that g''(x)>0 => g' strictly increasing, but with not luck. for b) I tried applying Bolzano theorem in [0,1] at the function h(x) = g(x) + 2 It is continuous because it is differentiative h(0) = g(0) + 2 h(1) = g(1) + 2 Then I noticed that on (2) in the limits of the integral there are g(0) and g(1) so if s(x) = image 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:image + but again I couldn't prove that f(0) = f(1) to apply the theorem. Any ideas?