Find the extreme values of f(x,y,z) = xy +xz on the curve of intersection of the right circular cylinder x^2+ y^2 = 1 and the hyperbolic cylinder xz =1.

Question

If I could get a worked out step by step solution for this it would really help to see it worked out.

Answer 1

Hi Mandy,

 

You can solve this problem similarly to the ellipsoid problem I posted an answer to earlier, using Lagrange multipliers. In this case, you're trying to optimize f(x,y,z) = xy + xz with two constraints g(x,y,z) = x² + y² = 1 and h(x,y,z) = xz = 1. Your Lagrange equation is then

∇f = λ ∇g + μ ∇h

 

Again, the partial derivatives are fairly simple, and you get the three equations

 

∂x: y + z = 2x λ + z μ

∂y: x = 2y λ + 0 μ

∂z: x = 0 λ + x μ

 

From the third equation, either x = 0 or μ = 1. x ≠ 0 since xz = 1. Therefore μ = 1.

 

The first equation then simplifies to

y + z = 2x λ + z

y = 2x λ

Substitute this into the second equation x = 2y λ in place of y, and you get

x = 2(2x λ) λ

x = 4x λ²

Since x≠0, divide by x and

1 = 4 λ²

λ = ±1/2

 

Plug these values into either the first or second equations and you find that either

x = y or x = -y (and in both cases, x² = y²)

 

Since x² + y² = 1

2 x² = 1

x = ±1/√(2)

y = ±1/√(2)

 

The final condition is xz = 1. Plug in each value for x, and you get

z = ±√(2) where x and z are either both positive or both negative.

 

These combine to give 4 critical values to test

(1/√(2), 1/√(2), √(2))
(1/√(2), -1/√(2), √(2))

(-1/√(2), 1/√(2), -√(2))
(-1/√(2), -1/√(2), -√(2))

 

Plug each of these into f(x,y,z) to determine which are the maximum and which are the minimum values.