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.