Lagrange multipliers with inequalities

Question

I have to find the max and min values of $x^2 + y^2 + xy$ bounded by $x^2 + y^2 \le 4$. I know how to do Lagrange multipliers, and have the points 0,0, x=y, -x=y, and -y=x, but I don't know how to incorporate the less than or equals sign to make sure I have all the points. Also, can someone please show me a laTex guide so I can make my equations look better I have no idea how to.

Answer 1

Try plotting the functions. $x^2 + y^2 \leq 4$ is a disc on the Euclidean plane. Plot $x^2 + y^2 + xy$ and see what you get.

Answer 2

One approach is to use Lagrange multipliers to find the extrema on the curve $x^2+y^2=4$, and separately, to find the critical points in the open set $x^2+y^2 < 4$ by finding the zeroes of the derivative.

Answer 3

Let $f(x) = x_1^2+x_2^2+x_1 x_2$

$C = \{ x | x_1^2+x_2^2 \le 4 \} $ is compact, so you know that $f$ has at least one maximizer and minimizer in $C$.

The set $C$ can be written as $C= C_i \cup C_b$, where $C_i = \{ x | x_1^2+x_2^2 < 4 \}$ and $C_b= \{ x | x_1^2+x_2^2 = 4 \}$.

Suppose $x$ is either a maximizer or minimizer.

If $x \in C_i$, then since $C_i$ is open, we must have ${\partial f(x) \over \partial x} = 0$. In particular, this implies that $x = 0$.

If $x \in C_b$, then the Lagrange approach gives (after working through the details) the possibilities $x = \pm \sqrt{2}, y = \pm \sqrt{2}$.

So, there are 5 points to be checked, the ones with the largest $f$ will be the maximizers and the ones with the lowest $f$ will be the minimizers.