Question

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)

Answer 1

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

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

$H=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}2&1\\1&2\end{bmatrix}\implies\det H=3>0$

Now since $\det H>0$ for any $(x,y)$, and $f_xx=2>0$ for any $(x,y)$, it follows by the second partial derivative test that the critical point $\left(\dfrac13,-\dfrac23\right)$ is the site of a local minimum of $f(x,y)$. The value of the minimum is $f\left(\dfrac13,-\dfrac23\right)=-\dfrac13$.