# Thread: Lagrange multipliers to find max and min

1. ## Lagrange multipliers to find max and min

The problem statement, all variables and given/known data

The temperature at a point (x, y) on a metal plate is T(x, y) = 4x^2 − 4xy + y^2 .
An ant, walking on the plate, traverses a circle of radius 5 centered at the origin.
Using the method of Lagrange multipliers, find the highest and lowest
temperatures encountered by the ant.

The attempt at a solution

T(x,y) = 4x^2 − 4xy + y^2
gradient of T = (8x - 4y)i + (2y - 4x)j

g(x,y) = x^2 + y^2 = 5^2
gradient of g = (2x)i + (2y)j

gradient of T = (lambda)gradient of g ----> lambda=#

8x - 4y = #2x ---->1
2y - 4x = #2y ---->2

# = 4-4y = 1-4x

what am i going to do next?

2. can someone give me and idea to proceed?

3. See attachment

4. i get the max is 45 and the min is 5..
am i calculated it right?

5. The max is 125 and the min is 0

max occur at [-2sqrt(5) ,sqrt(5)] and [2sqrt(5),-sqrt(5)]

min occur at [sqrt(5),2sqrt(5)] and [-sqrt(5),-2sqrt(5)]