I have this function :

$\displaystyle f(x,y)=x^2 + y^2 - 4x^{\color{red}2}y$

I should find the critical point.

http://en.wikipedia.org/wiki/Critical_point_(mathematics)
the system of first derivative :

$\displaystyle \left\{\begin{matrix} 2x-8xy=0 \\ 2y-4x^2=0 \end{matrix}\right. \quad$

solutions : $\displaystyle \quad p1(0,0) \quad p2(\frac{1}{\sqrt{8}}, \frac{1}{4}) \quad p3(\frac{-1}{\sqrt{8}},\frac{1}{4})$

then I should study the hessian matrix to understand the type of points

The determinant of Hessian is :

$\displaystyle H(x,y)=(2-8y)*2+8x \quad$

(where $\displaystyle f''_{xx}(x,y)=2-8y$ )

in $\displaystyle p1(0,0) \quad f''_{xx}(0,0)=2 \quad H(0,0)=4 \quad$ this is a point of Minimun

$\displaystyle p2(\frac{1}{\sqrt{8}}, \frac{1}{4}) \quad f''_{xx}(\frac{1}{\sqrt{8}}, \frac{1}{4})=0 \quad H(\frac{1}{\sqrt{8}}, \frac{1}{4})=\frac{8}{\sqrt{8}}$

$\displaystyle p3(\frac{-1}{\sqrt{8}},\frac{1}{4}) \quad f''_{xx}(\frac{-1}{\sqrt{8}},\frac{1}{4})=0 \quad H(\frac{-1}{\sqrt{8}},\frac{1}{4})=\frac{-1}{\sqrt{8}}$

The question :

P2 and P3 are points of saddle ? (Crying)