3x^3+6x^2-12x+8 - Wolfram|Alpha

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Input

3 x^3 + 6 x^2 - 12 x + 8

Plot

Plot

Plot

Alternate form

3 x (x (x + 2) - 4) + 8

x (x (3 x + 6) - 12) + 8

3 (x + 2)^3 - 12 (x + 2)^2 + 32

Real root

x = -2/3 (1 + 2^(1/3))^2

Complex roots

x = -2/3 + (2 2^(1/3))/3 + 2^(2/3)/3 + i (2^(2/3)/sqrt(3) - (2 2^(1/3))/sqrt(3))

x = -2/3 + (2 2^(1/3))/3 + 2^(2/3)/3 + i ((2 2^(1/3))/sqrt(3) - 2^(2/3)/sqrt(3))

Roots in the complex plane

Roots in the complex plane

Polynomial discriminant

Δ = -27648

Properties as a real function

Domain

R (all real numbers)

Range

R (all real numbers)

Surjectivity

surjective onto R

Derivative

d/dx(3 x^3 + 6 x^2 - 12 x + 8) = 3 (3 x^2 + 4 x - 4)

Indefinite integral

integral(3 x^3 + 6 x^2 - 12 x + 8) dx = (3 x^4)/4 + 2 x^3 - 6 x^2 + 8 x + constant

Local minimum

$min{3 x^3 + 6 x^2 - 12 x + 8} = 32/9 at x = 2/3$

Local maximum

$max{3 x^3 + 6 x^2 - 12 x + 8} = 32 at x = -2$

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