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

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

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Answer 1 · 2015-06-24T16:23:31

(0,0), ( $\sqrt{2}$ $sqrt2$ ,-1) and (- $\sqrt{2}$ $sqrt2$ ,-1)

Explanation:

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

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

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

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

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

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Explanation:

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How do you find the critical points of the function f(x,y)=x2+y2+(x2)(y)+4f(x,y)=x^2+y^2+(x^2)(y)+4?

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Explanation:

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