Hottest and Coldest Points on a Plate

A heated plate has a hottest and a coldest point somewhere. They can hide inside the plate, where the temperature levels off, or sit right on its edge, even in a corner. To find them, check both places.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Absolute extrema on a closed region

In one sentence

A continuous function on a closed and bounded region attains an absolute maximum and minimum there (Extreme Value Theorem); each occurs at an interior critical point or on the boundary.

General formula

\text{On a closed, bounded } {\htmlData{sym=1}{R}}:\quad \max_{\htmlData{sym=1}{R}} {\htmlData{sym=0}{T}} \text{ and } \min_{\htmlData{sym=1}{R}} {\htmlData{sym=0}{T}} \text{ occur where } {\htmlData{sym=2}{\nabla T}} = \mathbf 0 \text{ or on the boundary of } {\htmlData{sym=1}{R}}

Inside the plate a highest point must be flat, or you could climb higher. On the edge you are fenced in, so a corner or an edge point can win without being flat.

What the symbols mean

$T$
Temperature
How much warmer than the room each point of the plate is.
$R$
The plate
A closed, bounded region: it includes its edges.
$\nabla T$
Gradient
Zero exactly where the temperature levels off inside the plate.

Inside: solve $\nabla T = \mathbf 0$ . For $T = x^2 - xy + y^2 + bx$ that gives one point, $\left(-\tfrac{2b}{3},\ -\tfrac{b}{3}\right)$ .
On the boundary: each edge is a one-variable problem; check its critical points and the corners.
Compare every candidate value: the largest is the absolute maximum and the smallest the absolute minimum.
The Extreme Value Theorem promises both exist, because the plate is closed and bounded and T is continuous.

Learn the math

Absolute Minimums and MaximumsCalc III · Lesson 28, Applications of Partial Derivatives