1. ## multivariable extrema

hello!

i am having trouble with this question:

find the absolute extrema of $f(x,y) = 16+4y-2y^2+4x-2x^2$ defined on a closed region R bounded by the curve $x^2+y^2=4$

the question mentions that we can view f as a function of some t, for $0. Not sure how that helps, however.

to find extrma, we must check the critical points and the endpoints. i know a process for findings the critical points, which may be extrema or saddle points (when partials with respect to x and y both equal to zero). but not sure how to check extrema at endpoints?(the closed circle with radius two)

any help appriciated!
thanks.

2. Originally Posted by matt.qmar
hello!

i am having trouble with this question:

find the absolute extrema of $f(x,y) = 16+4y-2y^2+4x-2x^2$ defined on a closed region R bounded by the curve $x^2+y^2=4$

the question mentions that we can view f as a function of some t, for $0. Not sure how that helps, however.

to find extrma, we must check the critical points and the endpoints. i know a process for findings the critical points, which may be extrema or saddle points (when partials with respect to x and y both equal to zero). but not sure how to check extrema at endpoints?(the closed circle with radius two)

any help appriciated!
thanks.
This is an question about the extrema of a two variable function on a disk.
1- you should find the value of function at the critical points.
2- and you should find the maximum and the minimum on the boundary of the region.

Use Lagrange Multipliers to find the maximum and the minimum on the boundary of the region.

on the boundary, you want to find the maximum and the minimum of the function subject to the constraint $x^2+y^2=4$.

The Largest value from steps 1 and 2 is the maximum value and the smallest of these values is the minimum value.