Classsify-the-critical-points-of-the-function-f-x-y-x-2-y-1-3-y-3-x-2-y-2-2-Thank-you-in-advance-

Question Number 203858 by Mastermind last updated on 30/Jan/24

Classsify the critical points of the function

f (x, y) = x^{2} y + \frac{1}{3} y^{3} - x^{2} - y^{2} + 2

Thank you in advance!

Commented by TonyCWX08 last updated on 28/Feb/24

f_{x} (x, y) = 0

f_{x} (x, y) = 2 x y - 2 x

2 x y - 2 x = 0 \Rightarrow 1

f_{y} (x, y) = 0

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

x^{2} + y^{2} - 2 y = 0 \Rightarrow 2

S o l v i n g S i m u l t a n e o u s E q u a t i o n s

F r o m 1, 2 x y = 2 x \Rightarrow y = 1

S u b s t i t u t e y = 1 i n t o 2

x^{2} + 1 - 2 = 0

x^{2} = 1

x = 1 o r - 1

C r i t i c a l P o i n t s = (1, 1) o r (- 1, 1)