Question

12x³ y³

B- -4ax⁶ y²​

Answer 1

Respuesta:

A) Roots:

x = 0

y = 0

Polynomial discriminant:

Δ = 0

Properties as a function:

Domain

R^2

Range

R (all real numbers)

Parity

even

Partial derivatives:

d/dx(12 x^3 y^3) = 36 x^2 y^3

d/dy(12 x^3 y^3) = 36 x^3 y^2

Indefinite integral:

integral12 x^3 y^3 dx = 3 x^4 y^3 + constant

Definite integral over a disk of radius R:

integral integral_(x^2 + y^2<R^2) 12 x^3 y^3 dx dy = 0

Definite integral over a square of edge length 2 L:

integral_(-L)^L integral_(-L)^L 12 x^3 y^3 dy dx = 0

B) Result:

4 a x^6 y^2 + B

Roots:

a x!=0, y = -(i sqrt(B))/(2 sqrt(a) x^3)

a x!=0, y = (i sqrt(B))/(2 sqrt(a) x^3)

B = 0, a!=0, x = 0

Root:

a = 0, B = 0

Polynomial discriminant:

Δ_x = -47775744 a^5 B^5 y^10

Properties as a function:

Domain

R^2

Range

{z element R : B = z or (a!=0 and a B<=a z)}

Parity

even

Derivative:

d/dx(B - (-4 a x^6) y^2) = 24 a x^5 y^2

Indefinite integral:

integral(B + 4 a x^6 y^2) dx = 4/7 a x^7 y^2 + B x + constant

Definite integral over a hypercube of edge length 2 L:

integral_(-L)^L integral_(-L)^L integral_(-L)^L integral_(-L)^L (B + 4 a x^6 y^2) dy dx dB da = 0

Espero te sirva :)