Let

G = x^{3} e^{- x^{2} - y^{2}}

and

a = 3/2

G_{x} = x^{2} (3 - 2 x^{2}) e^{- x^{2} - y^{2}}, G_{y} = - 2 y x^{3} e^{- x^{2} - y^{2}},

so both vanish at

(a, 0)

and

(- a, 0)

. Evaluating the second derivatives at a point

(x, 0)

with

x^{2} = 3/2

gives

G_{xx} = - 6 x e^{- 3/2}, G_{x y} = 0, G_{yy} = - 3 x e^{- 3/2} .

18 x^{2} e^{- 3} = 27 e^{- 3} > 0

. Its two diagonal entries are negative at

x = a

and positive at

x = - a

. Hence

(a, 0) (- a, 0) : strict local maximum, : strict local minimum, G G = (\frac{3}{2 e})^{3/2}, = - (\frac{3}{2 e})^{3/2} .

These are also global extrema: for positive

x

, the factor

e^{- y^{2}} \leq 1

maximizes the positive value at

y = 0

, and differentiating

x^{3} e^{- x^{2}}

fixes its positive maximum at

a

. Oddness in

x

supplies the corresponding global minimum.

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