Let $\varphi (w)\Psi (x)$ be a solution to the heat problems. Given that $\Psi (-1)=\Psi (1)=0$ prove that it is not possible for $\Psi (x)$ and $\Psi ''(x)$ to be strictly positive for all $x$ such that $−1 < x < 1$. You may use any facts, such as the mean value theorem.
So yes, this is homework. But we have been at it for hours now,and could use a little nudge.
Assume $\Psi (x),\Psi ''(x)$ are positive. Then the mean value theorem states $\Psi '(c) = 0$ according to:
\begin{equation*} \Psi'(c) = \frac{\Psi (1)-\Psi (−1)}{1+(-1)}. \end{equation*}
Since $\Psi (c) = 0$, then we know there exists a tangent parallel to $x-$axis. This is the extrema which is strictly positive. Here is where we get stuck.
Since we assumed $\Psi ''(c)$ is positive then we know that there exists a minimum due to the the derivative test. But how does this help us prove that the range from $-1$ to $1$ is NOT strictly positive.
A little background info: This is for real analysis. Both me and roommate have not taken a prior proofs course (which is actually the pre-req). So we are not experts are proving so if any of you take the time to answer, please dumb it down as much as you can.