let-f-x-y-x-2-2xy-y-2-y-3-x-5-show-that-f-x-y-has-neither-a-maximum-nor-a-minimum-at-0-0-

Question Number 214805 by universe last updated on 20/Dec/24

let f (x, y) = x^{2} - 2 xy + y^{2} - y^{3} + x^{5} + show that

f (x, y) has neither a maximum nor a

minimum at (0, 0)

Answered by TonyCWX08 last updated on 20/Dec/24

f_{x} (x, y) = 2 x - 2 y + 5 x^{4} = 0

f_{y} (x, y) = - 2 x + 2 y - 3 y^{2} = 0

5 x^{4} + 2 x - 2 y = 0

- 2 x + 2 y - 3 y^{2} = 0

A d d i n g b o t h e q u a t i o n s g i v e

5 x^{4} - 3 y^{2} = 0

5 x^{4} = 3 y^{2}

y = x^{2} \sqrt{\frac{5}{3}}

5 x^{4} + 2 x - 2 (x^{2} \sqrt{\frac{5}{3}}) = 0

5 x^{4} + 2 x = 2 x^{2} \sqrt{\frac{5}{3}}

25 x^{8} + 20 x^{5} + 4 x^{2} = \frac{20}{3} x^{4}

75 x^{8} + 60 x^{5} + 12 x^{2} = 20 x^{4}

75 x^{8} + 60 x^{5} - 20 x^{4} + 12 x^{2} = 0

x^{2} (75 x^{4} + 60 x^{3} - 20 x^{2} + 12) = 0

x^{2} = 0

x = 0 \Rightarrow y = 0

A t (0, 0)

D (0, 0)

= f_{x x} (0, 0) f_{y y} (0, 0) - {[f_{x y} (0, 0)]}^{2}

= 0

N o t e n o u g h i n f o r m a t i o n t o p r o v e .