13/9/21 22:03
Taller 4-Hector Palencia
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Student: Hector Palencia
Date: 09/13/21
Instructor: Luis Eduardo Forero
Course: Análisis Multivariado y Funcional
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Assignment: Taller 4
The temperature at point (x,y) on a metal plate is . An ant on the plate walks around the circle of
radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?
T(x,y) = 576x − 576xy + 144y
2 2
First determine the equation for the circle centered at the origin with radius 5. This equation is shown below.
x y 25
2+2=
Suppose that f(x,y) and g(x,y) are differentiable. To find the local minimum of f subject to the constraint , find the values
of x, y, and that simultaneously satisfy the equations and . In this case,
and .
g(x,y) = 0
λ
∇
f =
λ
∇
g g(x,y) = 0
f(x,y) = T(x,y) = 576x − 576xy + 144y
2 2 g(x,y) = x + y − 25
2 2
Find .
∇
f
f(x,y) = 576x − 576xy + 144y
2 2
∇
f = (1152x − 576y) + (288y − 576x)i j
Find .
∇
g
g(x,y) = x y 25
2+2=
∇
g = (2x)i (2y)j+
Apply the Lagrange method in order to find the values of r, h, and .
λ
f
∇
= g
λ
∇
(1152x − 576y) + (288y − 576x)i j = ( i j )
λ
2x + 2y
First solve the equation i for .(1152x − 576y)i=
λ
2x
λ
(1152x − 576y)i=i
λ
2x
576 − 288y
x=
λ
Now solve the equation j for .(288y − 576x)j=
λ
2y
λ
(288y − 576x)j=j
λ
2y
144 − 288x
y=
λ
Set the two expression for equal to each other and solve for one variable in terms of the other.
λ
576 − 288y
x=144 − 288x
y
576xy − 288y2=144xy − 288x2Multiply both sides by xy.
288x + 432xy − 288y
2 2 = 0 Combine like terms on the left side.
144(2x − y)(x + 2y) = 0 Factor.
Now set each factor equal to 0 and solve.
2x y− = 0 or x 2y+ = 0
y = 2x x = 2y−
Thus, the local extrema occur when y 2x or when x 2y.= = −
