Solved: "Find the critical points of a function f(x,y)=xy2−3x2−y2−2x+2"

Monique Slaughter

Answered question

2021-12-16

Find the critical points of a function

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

Wendy Boykin

Beginner2021-12-17Added 35 answers

Partial derivatives of

z = f (x, y) = x y^{2} - 3 x^{2} - y^{2} - 2 x + 2

are

\frac{\partial z}{\partial x} = y^{2} - 6 x + 2

and

\frac{\partial z}{\partial y} = 2 x y - 2 y = 2 y (x - 1)

Make them equal to zero to find the critical points:

y^{2} - 6 x + 2 = 0, 2 y (x - 1) = 0

The second equation will be true when

y = 0

, then the first equation will be

- 6 x + 2 = 0

so that

6 x = 2

and

x = \frac{1}{3}

. Now, we have one critical point:

(x, y) = (\frac{1}{3}, 0)

The second equation wiil also be true when

x = 1

. Then, the first equation will be

y^{2} - 4 = 0

and

y^{2} = 4

, making

y = \pm 2

. Now, we have two critical points:

(x, y) = (1, 2)

and

(x, y) = (1, - 2)

Shannon Hodgkinson

Beginner2021-12-18Added 34 answers

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