x^4 - 4x^3 + 8x + 1 - Wolfram|Alpha

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Input

x^4 - 4 x^3 + 8 x + 1

Plot

Plot

Plot

Roots

x = 1 - sqrt(3 - sqrt(3))

x = 1 + sqrt(3 - sqrt(3))

x = 1 - sqrt(3 + sqrt(3))

x = 1 + sqrt(3 + sqrt(3))

Polynomial discriminant

Δ = 13824

Properties as a real function

Domain

R (all real numbers)

Range

{y element R : y>=-3}

Derivative

d/dx(x^4 - 4 x^3 + 8 x + 1) = 4 (x^3 - 3 x^2 + 2)

Indefinite integral

integral(x^4 - 4 x^3 + 8 x + 1) dx = x^5/5 - x^4 + 4 x^2 + x + constant

Global minima

$min{x^4 - 4 x^3 + 8 x + 1} = -3 at x = 1 + sqrt(3)$

$min{x^4 - 4 x^3 + 8 x + 1} = -3 at x = 1 - sqrt(3)$

Local maximum

$max{x^4 - 4 x^3 + 8 x + 1} = 6 at x = 1$

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