= Nonlinear ODE bounds from interior extrema
{title2=$f''=f+(f')^2$}
If a thrice differentiable solution has both endpoint values and both endpoint first derivatives bounded in modulus by $N$, it obeys $|f|\leq N$ and $|f'|\leq N$ throughout the interval. A positive interior maximum of $f$ would give $f''=f>0$, and a negative interior minimum would give $f''=f<0$, both impossible. For $w=f'$, differentiation gives $w''=w+2ww'$, so the same argument applies at an extremum of $w$. The resulting uniform bound and Lipschitz constant imply total boundedness of the solution family by the <Arzelà-Ascoli theorem>.
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