Lagrange Multipliers to find max/min values

arl146

1. The problem statement, all variables and given/known data
Use Lagange Multipliers to find the max and min values of the function subject to the given constraint(s). f(x,y)=exp(xy) ; constraint: x^3 + y^3 = 16


2. Relevant equations
[itex]\nabla[/itex]f = [itex]\nabla[/itex]g * [itex]\lambda[/itex]
f_x = g_x * [itex]\lambda[/itex]
f_y = g_y * [itex]\lambda[/itex]


3. The attempt at a solution
Set the fx and fy eqns equal to 0. but i cant solve for x, y, and lambda... i guess my algebra isnt that strong

i got f_x = [itex]\lambda[/itex] * g_x
y*e^xy = 3x²[itex]\lambda[/itex]

and for y:

x*e^xy = 3y²[itex]\lambda[/itex]

and g(x,y) = x³ + y³ = 16

arl146

no one can offer a little help? =/

I like Serena

Hi arl146!

You need to solve this set of equations:
(1) y e^xy = 3x²λ
(2) x e^xy = 3y²λ
(3) x³ + y³ = 16

Can you find λ from equation (1)?
And also from equation (2)?
Then equate them to each other, effectively eliminating λ?

HallsofIvy

Since a specific value for [itex]\lambda[/itex] is not necessary for the solution, I find it is often simplest to start by eliminating [itex]\lambda[/itex] by dividing one equation by another. Here, start by dividing [itex]ye^{xy}= 3x^2\lambda[/itex] by [itex]xe^{xy}= 3y^2\lambda[/itex]: [itex]y/x= x^2/y^2[/itex] which is the same as [itex]x^3= y^3[/itex]. Putting that into [itex]x^3+ y^3= 16[/itex] gives [itex]2x^3= 16[/itex].

arl146

well i sorta did something like that and got and y=x and with the whole x^3 + y^3 = 16 that means y=x=2. so theres what, is it called a critical pt still, at (2,2) ? so i just plug that into f(x,y)= exp(xy) ??
so you'd have f(2,2)=exp(4) .... is that all i do ? it just seems wrong.idk why haha

I like Serena

Yes that's all you do. :)

The only thing remaining is finding out whether it's a maximum or a minimum...

arl146

yea i think thats why i came here because i got confused with that one value. how do you know?

I like Serena

Doesn't your class material cover that?

Anyway, I know of 3 methods:

1. Using the second derivative test (Hessian matrix).
I can't quickly find a easy example for it (yet).

2. Since you only have one extrema, you can pick any point that satisfies the constraint and calculate f(x,y) there. Compare it with the f(x,y) at the extremum and you know whether it's a maximum or a minimum.

3. Vary x with a small epsilon, and calculate how you need to vary y to match the constraint in first order approximation.
In your case (x + epsilon)^3 + (y - epsilon)^3 = 16.
Check what f(x + epsilon, y - epsilon) does relative to f(x,y).