- #1

Yosty22

- 185

- 4

## Homework Statement

Find the shortest distance from the origin to the surface x=yz+10

## Homework Equations

## The Attempt at a Solution

So I said that my main function, f(x,y,z) = x^2 + y^2 + z^2 (the function I want to minimize)

Then I said that g(x,y,z) is my constraint function where g(x,y,z) is yz-x=-10. I took the partial derivative with respect to each variable of both g and f. I got f

_{x}=2x, f

_{y}=2y, f

_{z}=2z, g

_{x}=-1, g

_{y}=z, and g

_{z}=y. Once I did this, set f

_{x}= λ g

_{x}etc. (same format for each partial). This is where I am confused.

My final equations are:

2x + λ = 0 (1)

2y - λz = 0 (2)

2z - λy = 0 (3)

yz - x = 10 (4)

Once I have these, I am confused as to how to solve them properly. What I did so far was solve equation 2 for z. Once I solved for z in terms of y and λ, I substituted it back into equation 3 and got 4y/λ - λy = 0. Multiplying lambda across, I get 4y = λ

^{2}y. This shows me that either λ=2 or y=0. Once I get these, for each case I solved and when y = 0, plugging back into equation 2, I get z = 0, and this means that x=10 (equation 4). However, if λ = 2, then by equation 1, x=-1.

My question is:

What should I be looking for here? What do I solve for to answer the question properly?