How Do You Minimize xyz on the Unit Sphere Using Lagrange Multipliers?

[Ready2GoXtr]

Homework Statement

Minimize xyz on the unit sphere x2+y2+z2=1

Homework Equations

Lagrange Method.

The Attempt at a Solution

My attempt so far.. I am trying to follow the lagrange method

I set f(x,y,z)=xyz and g(x,y,z)=x²+y²+z²-1

g(x,y,z) = 0

I took the gradient of both functions

Δf(x,y,z) = (yz)i + (xz)j + (xy)k Δg(x,y,z) = (2x)i + (2y)j + (2z)k

Then set Δf(x,y,z) = λΔg(x,y,z)

Giving me the following

yz = λ2x
xz = λ2y
xy = λ2z

After that i am not sure what to do. Please help.

 

[gabbagabbahey]

Solve that system of 3 equations for x,y and z in terms of λ. Then use the constraint equation to determine λ and finally plug in your solutions to find the corresponding values of xyz...which ones are minimums? which are maximums?

 

[HallsofIvy]

Homework Statement

Minimize xyz on the unit sphere x2+y2+z2=1

Homework Equations

Lagrange Method.

The Attempt at a Solution

My attempt so far.. I am trying to follow the lagrange method

I set f(x,y,z)=xyz and g(x,y,z)=x²+y²+z²-1

g(x,y,z) = 0

I took the gradient of both functions

Δf(x,y,z) = (yz)i + (xz)j + (xy)k Δg(x,y,z) = (2x)i + (2y)j + (2z)k

Then set Δf(x,y,z) = λΔg(x,y,z)

Giving me the following

yz = λ2x
xz = λ2y
xy = λ2z

After that i am not sure what to do. Please help.

Since you are not interested in finding a value for $\lambda$, I recommend dividing one equation by another. For example dividing the first equation by the second gives y/x= x/y or x²= y² so y= x or y= -x. Similarly, dividing the first equation by the second gives z/x= x/z or x²= z² so z= x or z= -x. Put those into the condition that x²+ y²+ z²= 1 to determine specific values for x, y, and z.