The method of Lagrange multipliers

[yeland404]

Homework Statement

The problem of minimizing f(x1, x2) = x1^3
subject to
(x1 + 1)^3 = (x2 − 2)^2 is known to have a unique global solution. Use the method of Lagrange
multipliers to find it. You should deal with the issue of whether a constraint qualification holds.

Homework Equations

The problem of minimizing f(x) = 2x1^2+ x2^2+2x1*x2 -4x1-5x2+x3 subject to
2x1+x2+x3=0 is known to have a solution. Use the method of Lagrange
multipliers to find it. You should deal with the issue of whether a constraint qualification holds.

The Attempt at a Solution

I know for the queation be , the first step is to get the gradient of 2x1+x2+x3=0 then show is it nonzero everywhere, then use the Lagrange multiplier , but how about (x1 + 1)^3 = (x2 − 2)^2?

 

[vela]

First off, it doesn't make sense to say you're calculating the gradient of an equation. You calculate the gradient of a function.

You want to write the constraint in the form $g(x_1,x_2) = c$ where c is a constant. You then calculate $\nabla g$.

 

[HallsofIvy]

What, exactly, is the question? You first say "minimize $f(x_1,x_2)= x_1^3$" but under "relevant equations" you say "minimize $f(x)= 2x_1^2+ x_2^2+2x_1*x_2 -4x_1-5x_2+x_3$". Are these two separate problems? Also how are $x_1$, $x_2$, and $x^3$ connected with x? Did you mean $f(x_1, x_2, x_3)$?