Oxford Calculus: Classifying 2D Critical Points using the Discriminant

Video Description

University of Oxford Mathematician Dr Tom Crawford explains how to classify critical points for functions of two variables using the method of the discriminant. Maple Learn worksheet: https://learn.maplesoft.com/doc/rzlcu4rjpc/trm---critical-points-worksheet Just as the critical points for a function of one variable are found and classified using differentiation, the same techniques can be applied to a multivariable function to determine where it is stationary, and its behaviour near to the turning point. We begin with a description of the 3 types of critical point: maximum, minimum and saddle point with visualisations using the Maple Calculator app. You can download the app for free from Google Play and the App Store. Android: https://play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en Apple: https://apps.apple.com/us/app/maple-companion/id1466659419 Next, we look at how critical points are classified for functions of one variable, and then apply the same ideas to multivariable functions. Using the Taylor Expansion for a function of two variables, we derive an expression for the discriminant and explain why its sign determines the type of critical point present. You can follow all of the steps yourself using Maple Learn here: https://www.maplesoft.com/trm/maple-learn.aspx Finally, we apply the classification system using the method of the determinant to the example function seen in part 1. Part 1 on how to find critical points is here: https://youtu.be/Leomuu82-u8 Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: https://www.youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths