I've been struggling with this problem and I think that there may be an error in the book. The problem is:
If $\displaystyle f$ is a twice differentiable function with $\displaystyle f(0) = 0$ and $\displaystyle f(1) = 1$ and $\displaystyle f^{\prime}(0) = f^{\prime}(1) = 0$, then $\displaystyle |f^{\prime\prime}(x)| \geq 4$ for some $\displaystyle x \in [0,1]$.

If it were possible to use the fundamental theorem of Calculus, then I think I can do this. The problem is, at this point in the book all we have is the mean value theorem and its preceding theorems (boundedness, intermediate value theorem, etc.)

The other thing is, I looked up the solution in the manual which accompanies the book. I found that the solution was similar to the work I had done so far in that, assuming the theorem is not true, it is possible to show using the mean value theorem that $\displaystyle f(x) < 4x^{2}.$ But then the manual says 'it follows that $\displaystyle f(1/2) < 1/2$' whereas it actually only follows that $\displaystyle f(1/2)<1.$

For the proof to work, we would be required to show that $\displaystyle f(x) < 2x^{2}$ which I believe requires FTC, or maybe the problem should be to show that $\displaystyle |f^{\prime\prime}(x)| \geq 2$ for some $\displaystyle x \in [0,1].$

I know that the theorem is true, and I think it is probably quite a famous one (at least it ought to be!), but I don't know how to show it without FTC, and the error in the solutions manual makes me think that the question shouldn't be in this chapter, or that the author meant to write 2 instead of 4. What do you think?