# How do I determine the parameter a of the following equation such that f(x, y) has an extremum at a point (x, y) with x = -3, and show that it is a minimum?

## $f \left(x , y\right) = a {x}^{2} + x y + {y}^{2} + x + y + 1$

Mar 8, 2018

See below.

#### Explanation:

Calling

$p = \left(x , y\right)$
${p}_{0} = \left(- 3 , 1\right)$
$a = \frac{1}{3}$

we have

$f \left(x , y\right) = \left(p - {p}_{0}\right) \cdot A \cdot \left(p - {p}_{0}\right)$

with

$A = \left(\begin{matrix}\frac{1}{3} & \frac{1}{2} \\ \frac{1}{2} & 1\end{matrix}\right)$

$f \left(x , y\right)$ has a minimum at $p = {p}_{0}$