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I'm new to two variable calculus and having trouble classifying critical points for some functions which the second derivative test isn't really applicable.

E.g.

$$f(x,y) = x^2+y^2+x^2y+4 $$

$f_x=2x+2xy=0$

$f_y=2y+x^2=0$

Critical Point $(0,0)$

But I don't know how to determine the nature of this critical point.

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Express your objective function as: $$f(x,y)=x^2(1+y)+y^2+4.$$ Note that in the close neighborhood of $(0,0)$, the objective function is greater than or equal to $4$, hence $f(0,0)=4$ is the local minimum.

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Two more critical points are $(\sqrt 2,-1)$ and $(-\sqrt 2,-1)$. For finding the nature we use the hessian matrix as following $$H=\begin{bmatrix}2+2y&2x\\2x& 2y\end{bmatrix}$$The eigenvalues of $H$ for $(\pm \sqrt 2,-1)$ are $$\lambda (\lambda+2)-8=0\\\to \lambda_1=2,\lambda_2=-4$$therefore the points $(\pm \sqrt 2,-1)$ are saddle points. For $(0,0)$ we have $H\succeq 0$. Hence $(0,0)$ is a local minimum.

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  • $\begingroup$ Thank you. I fixed it.... $\endgroup$ – Mostafa Ayaz Nov 22 '18 at 7:19

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