Second derivative tests

One-variable second derivative test

Let $f:I\to\mathbb{R}$ be twice differentiable and let $a\in I^\circ$ satisfy $f'(a)=0$.

Proposition (one variable):

• If $f''(a)>0$, then $a$ is a strict local minimum of $f$.
• If $f''(a)<0$, then $a$ is a strict local maximum of $f$.
• If $f''(a)=0$, no conclusion follows in general.

Multivariable Hessian test

Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}$ be $C^2$ . Let $a\in U$ be a critical point , i.e. $$\\nabla f(a)$ =0$, and let $H=$\\nabla^2 f(a)$$ be the Hessian matrix at $a$.

Proposition (multivariable):

• If $H$ is positive definite (i.e., $v^{\mathsf T}Hv>0$ for all $v\neq 0$), then $a$ is a strict local minimum.
• If $H$ is negative definite (i.e., $v^{\mathsf T}Hv<0$ for all $v\neq 0$), then $a$ is a strict local maximum.
• If $H$ is indefinite (i.e., takes both positive and negative values on nonzero $v$), then $a$ is a saddle point (neither local min nor local max).
• If $H$ is only semidefinite, the test is inconclusive.

These tests are consequences of Taylor's theorem : near a critical point, the quadratic term $\frac12 h^{\mathsf T}Hh$ controls the leading behavior of $f(a+h)-f(a)$.

Proof sketch: One-variable: Taylor’s theorem gives $f(a+h)=f(a)+\tfrac12 f''(\xi)h^2$ for some $\xi$ between $a$ and $a+h$; the sign of $f''(a)$ controls the sign for $h$ small. Multivariable: Taylor’s theorem yields $f(a+h)=f(a)+\frac12 h^{\mathsf T}Hh+o(\|h\|^2).$ If $H$ is positive definite, then $h^{\mathsf T}Hh\ge c\|h\|^2$ for some $c>0$, which dominates the $o(\|h\|^2)$ remainder for small $h$, giving $f(a+h)>f(a)$ for $h\neq 0$ small.