- #1
smize
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Homework Statement
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.
f(x,y) = exy; g(x,y) = x3 + y3 = 16
Homework Equations
∇f(x,y) = λ∇g(x,y)
fx = λgx
fy = λgy
The Attempt at a Solution
∇f(x,y) = < yexy, xexy >
∇g(x,y) = < 3x2, 3y2 >
fx = λgx [itex]\Rightarrow[/itex] yexy = λ3x2
fy = λgy [itex]\Rightarrow[/itex] xexy = λ3y2
λ = [itex]\frac{xe^{xy}}{3y^{2}}[/itex] = [itex]\frac{ye^{xy}}{3x^{2}}[/itex] [itex]\Rightarrow[/itex] [itex]3x^{3}e^{xy} = 3y^{3}e^{xy}[/itex] [itex]\Rightarrow[/itex] x = y
Since x = y, [itex]x^{3} + y^{3} = 2x^{3} \Rightarrow 2x^{3} = 16, x = 2, y = 2[/itex]
f(2,2) = [itex]e^{(2)(2)} = e^{4}[/itex]
Here is where I'm having troubles. How do I determine whether it is a maximum or minimum? Could you give me more than one example? If I did anything wrong, please point it out.