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!


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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

$f_{x} (x, y) = 0$ $f_{x} (x, y) = 2 x y - 2 x$ $2 x y - 2 x = 0 \Rightarrow 1$ $f_{y} (x, y) = 0$ $f_{y} (x, y) = x^{2} + y^{2} - 2 y$ $x^{2} + y^{2} - 2 y = 0 \Rightarrow 2$ $S o l v i n g S i m u l t a n e o u s E q u a t i o n s$ $F r o m 1, 2 x y = 2 x \Rightarrow y = 1$ $S u b s t i t u t e y = 1 i n t o 2$ $x^{2} + 1 - 2 = 0$ $x^{2} = 1$ $x = 1 o r - 1$ $C r i t i c a l P o i n t s = (1, 1) o r (- 1, 1)$
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