Maximum principle states that U(x,t) takes maximum values only when at least one of the following holds: t=0 or x=0 or x=L. So your aim is to find where U(x,t) is at maximum. After you find the maximum point you will see that smth is wrong

P.S. When the author asks "where precisely the proof of maximum principle breaks down", he means that there is a standard way to prove the maximum principle for heat equation. And the question is "At which step

**exactly**, having this equation, we cannot continue moving, while we could have continued moving if we had heat equation.