Nonlinear ODE bounds from interior extrema ($f^{\prime \prime }=f+(f^{\prime }{)}^2$)

If a thrice differentiable solution has both endpoint values and both endpoint first derivatives bounded in modulus by $N$, it obeys $|f|\le N$ and $|f^{\prime }|\le N$ throughout the interval. A positive interior maximum of $f$ would give $f^{\prime \prime }=f>0$, and a negative interior minimum would give $f^{\prime \prime }=f<0$, both impossible. For $w=f^{\prime }$, differentiation gives $w^{\prime \prime }=w+2w{w}^{\prime }$, 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.