Local extrema

Exercises

1. Find the local extrema of the following functions

   • a) $f(x,y) = x^2+y^2$
   • b) $f(x,y) = x^2+xy+y^2-6x+6$

2. Find the local extrema of the function $f(x,y) = x^2+xy+y^2-6x+6$

3. Find the maximum and the minimum of the function $f(x, y) = x^2 + y^2 - x - y$ in the set $D = {(x, y) \in \mathbb{R}^2 : -1 \leq x \leq 1, -1 \leq y \leq 1}$.

4. Find the maximum and the minimum of the function $f(x, y) = x^2y$ in the set $D = {(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1}$.