# What are the extrema and saddle points of f(x, y) = x^2 + y^2+27xy+9x+3y?

Aug 26, 2016

A saddle point is located at $\left\{x = - \frac{63}{725} , y = - \frac{237}{725}\right\}$

#### Explanation:

The stationary poins are determined solving for $\left\{x , y\right\}$

$\nabla f \left(x , y\right) = \left(\begin{matrix}9 + 2 x + 27 y \\ 3 + 27 x + 2 y\end{matrix}\right) = \vec{0}$

obtaining the result

$\left\{x = - \frac{63}{725} , y = - \frac{237}{725}\right\}$

The qualification of this stationary point is done after observing the roots from the charasteristic polynomial associated to its Hessian matrix.

The Hessian matrix is obtained doing

$H = \nabla \left(\nabla f \left(x , y\right)\right) = \left(\begin{matrix}2 & 27 \\ 27 & 2\end{matrix}\right)$

with charasteristic polynomial

$p \left(\lambda\right) = {\lambda}^{2} - \text{trace} \left(H\right) \lambda + \det \left(H\right) = {\lambda}^{2} - 4 \lambda - 725$

Solving for $\lambda$ we obtain

$\lambda = \left\{- 25 , 29\right\}$ which are non zero with opposite sign characterizing a saddle point.