Absolute Maximum and Minimum Values of Multivariable Functions (Part 2 of 2)

In this video, I explain how to determine the absolute maximum and minimum values of a multivariable function on a closed, bounded set. You'll learn the full process of evaluating the function at its critical points and boundary values, using a clear example. The function we analyze is f(x,y)=3+xy−x−2y, evaluated over a triangular region with specified vertices.

What You Will Learn:

The step-by-step process to find critical points of a multivariable function.

How to analyze boundary values to identify extreme values.

A complete example of solving for the absolute max and min over a triangular region.

Practical tips for solving max/min problems in multivariable calculus.

How to use the given function f(x,y)=3+xy−x−2y for optimization.

This video is perfect for students learning multivariable calculus or those needing to review optimization techniques in calculus. If you find this video helpful, don’t forget to subscribe, leave a like, and share with your friends and classmates!

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[Icecube]

i want to add that you have to check the corners of the triangle as well. for this particular problem, the corners also give you 2, -2, and -2, so the corners didn't really matter. but other problems, like f(x, y) = 2 + 2x +2y -x^2 - y^2 in the 1st quadrant bounded by x = 0, y = 0, y = 4 -x will give you a max from one of the corners that you won't get if you just check the boundaries and the cp you get from setting fx and fy to 0 and solving for x and y.

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[AboDNayfeh]

Hi, why did you take the derivative of the third line and not for the first 2 lines ?