Mathematics: Given function f(x, y) = x 2 + xy + y 2 + y (1) locate critical points (2) classify those critical point (determine it’s local max or local min)

Given function f(x, y) = x 2 + xy + y 2 + y (1)

Ответ:

Boris1002

03.01.2022

Mathematics

$\nabla f(x,y)=(f_x,f_y)=(2x+y,x+2y+1)$

Set both partial derivatives equal to zero and solve for x,y

.

$\begin{cases}2x+y=0\\x+2y+1=0\end{cases}\implies x=\dfrac13,y=-\dfrac23$

The Hessian matrix (matrix of second-order partial derivatives) for this function is

$H=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}2&1\\1&2\end{bmatrix}\implies\det H=30$

Now since $\det H0$ for any (x,y)