# How do you find the critical points of the function f(x,y)=x^2+y^2+(x^2)(y)+4?

Jun 24, 2015

(0,0), ($\sqrt{2}$,-1) and (-$\sqrt{2}$,-1)

#### Explanation:

For the given function, ${f}_{x}$ = 2x +2xy and ${f}_{y} = 2 y + {x}^{2}$. Equate ${f}_{x}$and ${f}_{y}$ to 0 to get x(1+y)=0 and y= $- \left(\frac{1}{2}\right) {x}^{2}$. Solving these two

equations, we have $x \left(1 - \frac{1}{2} {x}^{2}\right) = 0.$ This gives x=0,$\sqrt{2} , - \sqrt{2}$ and

corresponding values of y =0,-1,-1.

The critical points are (0,0), ($\sqrt{2}$,-1) and (-$\sqrt{2}$,-1)