Multivariable Calculus Question >_<?
Let T(x,y,z)=20+2x+2y+z^2 represent the temperature at each point on the sphere x^2+y^2+z^2=11. Find the extreme temperatures on the curve formed by the intersection of the plane x+y+z=3 and the sphere.
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MAX(20+2x + 2y + z^2)
subject to the constraints
x^2 + y^2 + z^2 - 11 = 0
x + y + z - 3 = 0
F(x,y,z) = (20 + 2x + 2y + z^2) + lambda1*(x^2 + y^2 + z^2 - 11) + lambda2*(x+y+z-30
dF/dx = 2 + 2*lamdba1*x + lambda2 = 0
dF/dy = 2 + 2*lambda1*y + lambda2 = 0
dF/dz = 2z +2z*lambda1 + lambda2 = 0
dF/dlambda1 = x^2 + y^2 + z^2 - 11 = 0
dF/lambda2 = x+y+z-3 = 0
by the fourth equation and the fifth equation, we conjecture that the z= 3, x = -1, y = 1. This leads to lambda1 = 0 and lambda2 = -2, but it also implies that z = 1. contradiction.
next, we try the case, z = 1, x=3, y=-1. lambda1 = 0, lamdba2 = -2. By the third equation, z = 1, so it checks out.
One such extreme temperature is therefore,
T = 20 + 6 -2 + 1 = 25 degrees, and this is the maximum