Respuestas
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 :)
