Problem Zero: Read the section and make your own summary of the material.

We need to read the section $12.7$and have to make the summary of the section.

• The Method of Lagrange Multipliers:

Let $f$and $g$be functions with continuous first-order partial derivatives. If $f(x,y)$has a relative extremum at a point $\left(x_0,y_0\right)$subject to the constraint $g(x,y)=0$, then $\nabla f\left(x_0,y_0\right)$and $\nabla g\left(x_0,y_0\right)$are parallel. Thus, ifrole="math" localid="1650003963919" $\nabla g\left(x_0,y_0\right)\ne 0$.

$\nabla f\left(x_0,y_0\right)=\lambda \nabla g\left(x_0,y_0\right)$

for some scalar$\lambda$, where$\lambda$is called a Lagrange multiplier.

• The Extreme Value Theorem for a Function of Two Variables:

Let $f$be a continuous function of two variables defined on the closed and bounded set $S$. Then there exist points $\left(x_M,y_M\right)$and $\left(x_m,y_m\right)$in $S$such that role="math" localid="1650004575191" $f\left(x_M,y_M\right)$is the maximum value of $f$on $S$and $f\left(x_m,y_m\right)$is the minimum value of $f$on $S$.

• Optimizing a Function with Two Constraints:

Let us consider the constraints, $g(x,y,z)=0$and $h(x,y,z)=0$, define surfaces in${\mathrm{ℝ}}^3$. Assuming that these surfaces intersect in a curve, we would be attempting to find the maximum and minimum values of $f$on this curve of intersection. To do this, we attempt to solve the system given by,

$\nabla f\left(x_0,y_0,z_0\right)=\lambda \nabla h\left(x_0,y_0,z_0\right)+\mu \nabla h\left(x_0,y_0,z_0\right)$

where $\lambda ,\mu$are both Lagrange multipliers.