# Lagrang multipliers to find max and min

## Homework Statement

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

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

# = 4-4y = 1-4x

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lanedance
Homework Helper
haven't checked your work, but notice you also have the original constraint, use that with the equation that you found

haven't checked your work, but notice you also have the original constraint, use that with the equation that you found
what is the meaning of original constraint?

lanedance
Homework Helper
what is the meaning of original constraint?
you have a function to optimise T(x, y) = 4x^2 − 4xy + y^2, against a given constraint x^2 + y^2 = 5^2

if i solved the previous equation, i get, x=(1/4)y

i've got max= 45 and min=5..
am i done it right?

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HallsofIvy
Homework Helper
Yes, what you have done so far is correct. But that certainly does NOT give "max= 45 and min=5"!

ok...
when y = 2x,
i substitute into x^2+y^2=25
i got x=+-(5)^1/2

when x=-2y,
i substitute into x^2+y^2=25
i got y= +-(5)^1/2

so.. i got my critical points..
---> [(5^1/2) , (5^1/2)]
---> [(5^1/2) , -(5^1/2)]
---> [-(5^1/2) , (5^1/2)]
---> [-(5^1/2) , -(5^1/2)]

my critical points are correct?:uhh: