Lagrange multipliers finding the maximum and minimum.

Question

Consider the two-variable function $f(x, y) = x^ 2 + 2y^2$

(a) Find the maxima and minima of $f(x, y)$ on the unit circle $x^2 +y^2 = 1$

I have used lagrange multipies to get landa= to $1$ or $2$. Given the points$(1,0),(-1,0),(0,1),(0,-1)$. so the min would be at $(0,0)$ and the max at $(0,-1)$ and $(0,1)$

Am i right in thinking this?

Secondandly how can i sketch a contour plot of this question to show graphically mypoints of maximum and minimum. I am aware its an ellipse contour.

thanks

Answer 1

For that case we have:

$$f(x,y)=x^2+y^2+y^2=1+y^2$$

because we are looking for minumim and maximum using the constrain $x^2+y^2=1$.

But $y^2=1-x^2$ so the maximum of $y^2$ is $1$ (when $x=0$) and the minimum is $0$ (when $x=\pm1$).

So

$$f_{max}=1+1=2\\ f_{min}=1+0=1$$