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

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Answer 1 · 2016-08-26T13:23:34

A saddle point is located at ${x = -63/725, y = -237/725}$

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

$grad f(x,y) = ((9 + 2 x + 27 y),( 3 + 27 x + 2 y)) = vec 0$

obtaining the result

${x = -63/725, y = -237/725}$

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

$H = grad(grad f(x,y)) = ((2,27),(27,2))$

with charasteristic polynomial

$p(lambda) = lambda^2- "trace"(H)lambda + det(H) = lambda^2-4 lambda-725$

Solving for $lambda$ we obtain

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

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