Lecture 12     

Local Min/Max

APMA E2000

Drew Youngren dcy2@columbia.edu

$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$ $\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$ $\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$

Announcements

  • Quiz 4 (on HW 6) this week
    • Chain Rule
    • Directional Derivatives
    • Properties of the Gradient
  • HW7 due Tues

1-minute review

The directional derivative of a differentiable function $f$ in the direction of unit vector $\vec u$ at $\vec x$ is \[D_{\vec u} f(\vec x) = \lim_{h\to 0}\frac{f(\vec x+h\vec u) - f(\vec x)}{h} = \nabla f(\vec x)\cdot\vec u.\]

3 important properties:

  • $\nabla f$ is orthogonal to level sets.
  • $\nabla f$ points in the direction of greatest change.
  • $|\nabla f|$ is the amount of greatest change.

$\displaystyle \nabla f = \vec 0$

Definition

A local minimum (resp. maximum) of a scalar field $f$ is a value $f(\vec a)$ such that \[f(\vec a) \leq f(\vec x)\] (resp. $f(\vec a) \geq f(\vec x)$) for all positions $\vec x$ in some open set containing $\vec a$.

Definition

An absolute minimum (resp. maximum) of a scalar field $f$ on a particular domain $D$ is a value $f(\vec a)$ such that \[f(\vec a) \leq f(\vec x)\] (resp. $f(\vec a) \geq f(\vec x)$) for all positions $\vec x \in D$.

Main Examples

Consider functions of the form \[ f(x,y) = \pm x^2 \pm\ y^2. \]

Definition

A critical point of a function $f$ is a position $a$ such that $f$ is not differentiable or \[\nabla f(\vec a) = \vec 0. \]

Main Result

Suppose $f(\vec a)$ is a local extremum, then $\vec a$ is a critical point.

Example

Identify the critical points from the contour plot below. Do they correspond with a local min or max or neither?

Example

Find all critical points of the function \[ f(x,y) = x^4 + y^4 + 4 x y - 1.\]

Solution. \[\nabla f = \bv{4x^3 + 4y \\ 4 y^3 + 4x} = \bv{0 \\ 0}\]

Substituting $y = -x^3$,\[ x^9 - x = \]\[ x(x - 1)(x + 1)(x^2 + 1)(x^4 +1) = 0 \] has roots $x = 0, \pm 1$.

This gives 3 critical points: $(-1,1)$, $(0,0)$, and $(1, -1)$.

Classifying Critical Points

Recall

In 1-D, the second-derivative determines classification at critical points.

A graph with a min a max and a saddle point in 1-D.
The Second Derivative Test
If all 2nd order partials of $f(x,y)$ are continuous in the neighborhood of a critical point $(a,b)$, let \[ D = f_{xx}f_{yy} - f_{xy}f_{yx} = \begin{vmatrix} \frac{\partial ^2 f}{\partial x^2} & \frac{\partial ^2 f}{\partial y \partial x} \\ \frac{\partial ^2 f}{\partial x \partial y} & \frac{\partial ^2 f}{\partial y^2} \end{vmatrix}.\]
  • if $D>0$ and $f_{xx} < 0$, $f(a,b)$ is a local maximum.
  • if $D>0$ and $f_{xx} > 0$, $f(a,b)$ is a local minimum.
  • if $D<0$, $(a,b)$ is a saddle point.

Otherwise, the test is inconclusive.

Example

Classify the critical points of the function \[ f(x,y) = x^4 + y^4 + 4xy - 1 \] above.

$(x,y)$ $f_{xx}$ $f_{yy}$ $f_{xy}$ $D$ class
$(0,0)$ 0 0 4 -16 saddle
$(1,-1)$ 12 12 4 128 min
$(-1,1)$ 12 12 4 128 min

graph

Optimization

Some Topology

An open set $U\subset \RR^n$ is one where each element can be surrounded by a (small) ball of elements in the set.

A closed set contains all its boundary points.

Illustration of niterior point and boundary point
Illustration of sets that are closed, open, and neither.

Kinds of Optimization

Unconstrained

On open sets $\longrightarrow$ look for critical points

Constrained

On boundary points $\longrightarrow$ Lagrange multipliers

Example - Unconstrained

Find the closest point to the origin on the plane \[z = x -2y + 3.\]

Learning Outcomes

You should be able to...
  • Provide a technical definition of local extrema.
  • Identify critical points of scalar fields.
  • Give examples of each class of critical point in 2 and 3 dimensions.
  • Correctly implement the 2nd derivative test in 2 dimensions.