How to use the Lagrange Multipliers to find the min and max of this function?

Question

So I have the function $$f(x,y)=x^2+y^2$$ with constraint $$(x-1)^2+4y^2=4$$

How can I find the minimum and maximum values for this using Lagrange multipliers?

My attempt: I got the equations:

1) $$2x = λ(2x-2)$$ 2) $$2y = λ8y$$ 3) $$(x-1)^2+4y^2=4$$

I solved for x and y with no luck. Cant seem to find the min and max. Any help is appreciated

Answer 1

You are on the right track!

The first thing to notice is that the constraint is bounded. Since $f$ is a continuous function, Weierstrass theorem guarantees the existence of a maximum and a minimum.

• If $\lambda = 0$ then, by $(1)$ and $(2)$, necessarily $x = y = 0$. Since these values don't satisfy $(3)$ we conclude that $\lambda \neq 0$.
• If $y = 0$ then from $(3)$ we get $x = 3$ or $x = -1$.
• If $y \neq 0$ then we can find $\lambda$ from $(2)$: $\lambda = \frac 14$. Plug this value into $(1)$ and solve for $x$ to get $x = -\frac 13.$ Plug this value into $(3)$ to find $y$. You should get $y = \pm\frac{\sqrt{5}}{3}$.

This covers all the possibilities and after this procedure we are left with the following "interesting" points: $$(-1,0), (3,0), (-\frac 13, \frac{\sqrt{5}}{3}), (-\frac 13, -\frac{\sqrt{5}}{3}).$$

Now evaluate $f$ at these points and draw your conclusions! :)