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?