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

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Answer 1 · 2015-06-24T16:23:31

(0,0), ( $sqrt2$ ,-1) and (- $sqrt2$ ,-1)

For the given function, $f_x$ = 2x +2xy and $f_y= 2y +x^2$ . Equate $f_x$ and $f_y$ to 0 to get x(1+y)=0 and y= $-(1/2) x^2$ . Solving these two

equations, we have $x(1-1/2 x^2)=0.$ This gives x=0, $sqrt 2, -sqrt2$ and

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

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

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