334-The 2nd Derivative test.sagews

The 2nd Derivative test

The second derivative test looks at the eigenvalues of the Hessian matrix evaluated at critical points.

The idea for this came from Ben Woodruff, who implemented a similar thing in Mathematica.

var('x,y')

(x, y)

f(x,y)=-x^2+y^2
table, graph=illustrate_test(f)
print table
show(graph)

Critical Point	Eigenvalue	Eigenvector
(0.0, 0.0, 0.0)	-2.0	(1.0, 0.0)
(0.0, 0.0, 0.0)	2.0	(0.0, 1.0)

f(x,y)=x*y*exp((-x^2-y^2)/3)
table, graph=illustrate_test(f)
print table
show(graph)

Critical Point	Eigenvalue	Eigenvector
(0.0, 0.0, 0.0)	1.0	(0.707106781187, 0.707106781187)
(0.0, 0.0, 0.0)	-1.0	(-0.707106781187, 0.707106781187)
(-1.22474487139, -1.22474487139, 0.551819161757)	-0.735758882343	(0.707106781187, 0.707106781187)
(-1.22474487139, -1.22474487139, 0.551819161757)	-0.735758882343	(-0.707106781187, 0.707106781187)
(1.22474487139, -1.22474487139, -0.551819161757)	0.735758882343	(0.707106781187, 0.707106781187)
(1.22474487139, -1.22474487139, -0.551819161757)	0.735758882343	(-0.707106781187, 0.707106781187)
(-1.22474487139, 1.22474487139, -0.551819161757)	0.735758882343	(0.707106781187, 0.707106781187)
(-1.22474487139, 1.22474487139, -0.551819161757)	0.735758882343	(-0.707106781187, 0.707106781187)
(1.22474487139, 1.22474487139, 0.551819161757)	-0.735758882343	(0.707106781187, 0.707106781187)
(1.22474487139, 1.22474487139, 0.551819161757)	-0.735758882343	(-0.707106781187, 0.707106781187)

illustrate_test?