### Multivariable Optimization using Lagrange Multipliers Example 2

We can use Lagrange multipliers to solve problems where we're asked to find the max/min of an objective function, subject to equation constraints.

Find the extreme values of $$f(x,y,z) = x + 2y$$ subject to both constraints $$x + y + z = 1$$ and $${y^2} + {z^2} = 4$$.

For Lagrange multipliers involving 2 constraint equations, we can use:

$\nabla f = \lambda \nabla g + \mu \nabla h$

Where lambda and mu are unknown values. We can also set the components of the gradients equal individually, so we can obtain 3 equations, each one with the variables x, y and z. For now I will use vector notation and I'll fill in the gradients.

$\left\langle {1,2,0} \right\rangle = \lambda \left\langle {1,1,1} \right\rangle + \mu \left\langle {0,2y,2z} \right\rangle$

In this vector equation, as I said earlier you can set each component equal to create three equations. Looking at just the first component,

$1 = \lambda (1) + \mu (0)$

$\lambda = 1$

Now use the y and z components from our vector equations and solve.

$2 = (1)(1) + \mu (2y)$

$0 = (1)(1) + \mu (2z)$

From these two equations, we find that:

$y = \frac{1}{2}\mu$

$z = - \frac{1}{2}\mu$

Or in other words,

$y = - z$

Then plug into the constraint $${y^2} + {z^2} = 4$$.

${( - z)^2} + {z^2} = 4$

${z^2} = 2$

$z = \pm \sqrt 2$

And we know that y is just negative z. I flip the plus or minus sign to a minus or plus sign for y.

$y = \mp \sqrt 2$

Now plug these values into the other constraint, $$x + y + z = 1$$.

$x + \sqrt 2 - \sqrt 2 = 1$

$x = 1$

And the other case,

$x - \sqrt 2 + \sqrt 2 = 1$

$x = 1$

So the two critical points are $$f(1,\sqrt 2 , - \sqrt 2 )$$ and $$f(1, - \sqrt 2 ,\sqrt 2 )$$. Then just plug these points into our objective function, $$f(x,y,z) = x + 2y$$, and see which point is the max and which is the min.

$f(1,\sqrt 2 , - \sqrt 2 ) = 1 + 2\sqrt 2$

$f(1, - \sqrt 2 ,\sqrt 2 ) = 1 - 2\sqrt 2$

So the first point above is the max, while the second point above is the min.

Please see our other worked-out examples of multivariable optimization using either Lagrange multipliers or the second derivative test.

Lagrange Multiplier Examples:

Lagrange Multipliers Optimization Ex1

Lagrange Multipliers Optimization Ex3

Second Derivative Test Examples:

Second Derivative Test Optimization Ex1