2012-10-29, 12:41 AM
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.)
f(x, y) = 5x^2 + 5y^2; xy = 1
I'm confused. Normal way isn't working?
F(x,y,λ
= f(x,y) - λ(g(x,y)) - k
F(x,y,λ = 5x^2 + 5y^2 - λxy + λ
Fx = 10x - λy
Fy = 10y - λx
Fλ = -xy + 1
Setting derivatives = 0 and solving:
y = 10x/λ; x = 10y/λ; xy = 1
Sub y and x into xy = 1:
(10x/λ(10y/λ = 1
λ = 10sqrt(xy)
Sub λ back into fx = 0, fy = 0:
y = 10x/10sqrt(xy)
y^2 = x/y
y^3 = x
x = 10y/10sqrt(xy)
x^2 = y/x
x^3 = y
What do I do from here? I feel like I messed up somewhere in the process, because I don't know what to do with a variable.
f(x, y) = 5x^2 + 5y^2; xy = 1
I'm confused. Normal way isn't working?
F(x,y,λ
= f(x,y) - λ(g(x,y)) - kF(x,y,λ
= 5x^2 + 5y^2 - λxy + λFx = 10x - λy
Fy = 10y - λx
Fλ = -xy + 1
Setting derivatives = 0 and solving:
y = 10x/λ; x = 10y/λ; xy = 1
Sub y and x into xy = 1:
(10x/λ
(10y/λ = 1λ = 10sqrt(xy)
Sub λ back into fx = 0, fy = 0:
y = 10x/10sqrt(xy)
y^2 = x/y
y^3 = x
x = 10y/10sqrt(xy)
x^2 = y/x
x^3 = y
What do I do from here? I feel like I messed up somewhere in the process, because I don't know what to do with a variable.