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

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Multivariable Optimization

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