From 93386a36ffb709030249ab44b6c0b1505b6637d9 Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 24 Jan 2023 18:39:17 -0500 Subject: [PATCH] math119: add second derivative test --- docs/1b/math119.md | 14 ++++++++++++++ 1 file changed, 14 insertions(+) diff --git a/docs/1b/math119.md b/docs/1b/math119.md index ffe717b..892f05c 100644 --- a/docs/1b/math119.md +++ b/docs/1b/math119.md @@ -274,3 +274,17 @@ Cartesian and polar coordinates can be easily converted between: - $x=r\sin\theta\cos\phi$ - $y=r\sin\theta\sin\phi$ - $z=r\cos\theta$ + +## Optimisation + +**Local maxima / minima** exist at points where all points in a disk-like area around it do not pass that point. Practically, they must have $\nabla f=0$. + +**Critical points** are any point at which $\nabla f=0|undef$. A critical point that is not a local extrema is a **saddle point**. + +Local maxima tend to be **concave down** while local minima are **concave up**. This can be determined via the second derivative test. For the critical point $P_0$ of $f(x,y)$: + +1. Calculate $D(x,y)= f_{xx}f_{yy}-(f_{xy})^2$ +2. If it greater than zero, the point is an extremum + a. If $f_{xx}(P_0)<0$, the point is a maximum — otherwise it is a minimum +3. If it is less than zero, it is a saddle point — otherwise the test is inconclusive and you must use your eyeballs +