Extrema Question

Question

$f(x,y)=x^2+x\cdot y+x+y^2$ on the set $S=\{(x,y) : x^2+y^2 \leq 9\}$.

I found the critical point on the interior of $S$ to be $\left(\frac{-2}{3},\frac{1}{3}\right)$.

I'm not sure how to parametrize the boundary of $S$ in terms of theta using sine and cosine and write $f$ in terms of theta on the boundary by putting restrictions on theta.

I also am not sure how to find the extrema of $f$ on $S$.

Answer 1

Use $$\sin(\theta)=\frac{y}{r}$$ and $$\cos(\theta)=\frac{x}{r}$$ where $$r=\sqrt{x^2+y^2}$$