I am studying the heat transfer problem of a planar lamina and I have encountered the following fact:
Let $f(x)$ be a $C^2$-class function defined in $[-1,1]$. The book claims that, since the BCs are $f(-1)=f(1)=0$, then $f(x) $ and $ f''(x)$ cannot be both strictly positive (or negative) for any $x \in [-1,1]$ .
I have tried to apply the "theorem of sign permanence", but unsuccessfully. I hope for some hints. Thanks!
Since $f(-1)=f(1)=0$, by Mean Value theorem, there exists some point $x_0\in (-1,1), $ such that $f'(x_0)=0$. Next, we take Taylor series at $x=x_0$
$$\begin{align}f(-1)&=f(x_0)+f'(x_0)(-1-x_0)+\frac{f''(c)}{2}(-1-x_0)^2,~~~~~c\in (-1, x_0)\\ \\ 0&=f(x_0)+\frac{f''(c)}{2}(-1-x_0)^2\end{align}$$
Without loss of generality, assume $f(x)>0, f''(x)>0$, then we get contradictions,
$$0>0$$