Solve the optimization problem

Maximize P = xyz with x + y = 36 and y + z = 36, and x, y, and z ≥ 0

p = ??

Question

Solve the optimization problem

Maximize P = xyz with x + y = 36 and y + z = 36, and x, y, and z ≥ 0

p = ??

tkhunny · Answer

Shall we go with mostly algebra (simple substitution) or mostly calculus (Lagrange Multipliers)?

anonymous · Answer

calculus

tkhunny · Answer

That's annoying.  It's pretty tedious.  Are you sure yu don't want to solve the linear constraints for x and z and substitute the results into P?  Then we would have a single variable and we'd be almost done.

anonymous · Answer

i have to

tkhunny · Answer

Well, then get to it.  Create $f(x,y,z,\lambda,\mu) = xyz - \lambda(x+y-36) - \mu(y+z-36)$ and calculate five partial derivatives:  $$\dfrac{\partial{f}}{\partial{x}} = yz - \lambda$$  Let's see the rest.

Note: That x, y, z, > 0 restriction WILL help.