Mathematics, 07.04.2020 01:10 nadarius2017

2x2y + 8x3 – xy2 – 2x3 + 3xy2 + 6y3

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Answer from: Queey

2 x y^2 + 6 y^3 + 4 y + 18

Step-by-step explanation:

2×2 y + 8×3 - x y^2 - 2×3 + 3 x y^2 + 6 y^3

Result:

2 x y^2 + 6 y^3 + 4 y + 18

3D plot:

3D plot

Contour plot:

Contour plot

Alternate forms:

2 (x y^2 + 3 y^3 + 2 y + 9)

2 (y (y (x + 3 y) + 2) + 9)

y (y (2 x + 6 y) + 4) + 18

Real roots:

x≈-9.24225, y≈-0.797732

x≈-9.24225, y≈1.93924

Roots:

y = (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)/(9 2^(1/3)) - (2^(1/3) (18 - x^2))/(9 (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9

y = -((1 - i sqrt(3)) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3))/(18 2^(1/3)) + ((1 + i sqrt(3)) (18 - x^2))/(9 2^(2/3) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9

y = -((1 + i sqrt(3)) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3))/(18 2^(1/3)) + ((1 - i sqrt(3)) (18 - x^2))/(9 2^(2/3) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9

Polynomial discriminant:

Δ_y = -16 (36 x^3 - 4 x^2 - 972 x + 19779)

Integer roots:

x = -14, y = 1

x = -4, y = -1

Properties as a function:

Domain

R^2

Range

R (all real numbers)

Roots for the variable y:

Partial derivatives:

d/dx(2 x y^2 + 6 y^3 + 4 y + 18) = 2 y^2

d/dy(2 x y^2 + 6 y^3 + 4 y + 18) = 4 x y + 18 y^2 + 4

Indefinite integral:

integral(18 + 4 y + 2 x y^2 + 6 y^3) dx = x^2 y^2 + 6 x y^3 + 4 x y + 18 x + constant

Definite integral over a disk of radius R:

integral integral_(x^2 + y^2<R^2)(2 x y^2 + 6 y^3 + 4 y + 18) dx dy = 18 π R^2

Definite integral over a square of edge length 2 L:

integral_(-L)^L integral_(-L)^L (18 + 4 y + 2 x y^2 + 6 y^3) dy dx = 72 L^2

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a. since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

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