Find maxi,minimum $f(x,y)=x^3+y^3 (\text{where} ~~~ x,y\in \mathbb{R}, x^2+y^2=1)$ I would appreciate if somebody could help me with the following problem
Q: Find maximum and minimum of $f(x,y)$ that
$$f(x,y)=x^3+y^3 (\text{where} ~~~ x,y\in \mathbb{R}, x^2+y^2=1)$$

Find max:
$x^2 + y^2 = 1 \implies x, y\in[-1,1] \implies x^2 \ge x^3,  y^2\ge y^3 \implies 1 = x^2+y^2\ge x^3 + y^3$
Follow that, find min: It is easy to see that $x^3 + y^3$ min $\iff$ $x, y < 0$ and $|x^3 + y^3|$ max

Cleaner:
$$x^2 \ge |x|^3,  y^2\ge |y|^3 \implies 1= x^2 + y^2 \ge|x^3|+|y^3| \ge|x^3 +y^3|$$ 
$$\iff x^3+y^3 \in[-1,1]$$ 
