= Uniform negative curvature and global maximization
{title2=$f''(x)\le-c<0$}
If a twice differentiable real <function> on $\mathbb R$ satisfies $f''(x)\le-c$ for some fixed $c>0$, then it has a unique <global maximum>. The <mean value theorem> shows that $f'$ is strictly decreasing, and that $f'(x)+cx$ is nonincreasing. Hence $f'(x)\le f'(0)-cx$ for $x>0$ and $f'(x)\ge f'(0)-cx$ for $x<0$. These inequalities force opposite derivative signs at the two ends. The <intermediate value theorem> supplies a unique zero of $f'$, at which $f$ changes from increasing to decreasing. Merely having $f''<0$ does not guarantee existence: $f(x)=-e^{-x}$ is increasing and approaches its unattained supremum zero.
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