I am unable to understand how to use lagrange multipliers in this question. Can you lead me a way to solve this question? I've always used lagrange multipliers with circles; however, in this case, it is an ellipsis. I am unsure about how to apply lagrange multipliers on ellipsis.
Question: Function: $f(x,y)=\frac{1}{3}\cdot \:x^3-\frac{1}{4}\cdot \:x+y^2$
Find the absolute extreme values on the region $D$: $$D=\left\{\left(x,y\right)\in\mathbb{R}^2:\left(x-1\right)^2+4y^2\le 4 \right\}$$
First step: determine the critical points of $f$ in the interior of $D$ and investigate their nature (lokal max, local min, saddle point).
Second step: investigate the function $F$ on $ \partial D.$ The Lagrange function is:
$$L(x,y, \mu)=f(x,y) - \mu( (x-1)^2+4y^2-4).$$