f(x)=(x-1/5)e^(10x) - Wolfram|Alpha

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Input

f(x) = (x - 1/5) e^(10 x)

Plot

Plot

Plot

Expanded form

f(x) = e^(10 x) x - e^(10 x)/5

Alternate form

f(x) = 1/5 e^(10 x) (5 x - 1)

Logarithmic form

log(f(x)) = log(x - 1/5) + (10 x) log(e)

Root

x = 1/5

Properties as a real function

Domain

R (all real numbers)

Range

{f element R : f>=-e/10}

Series expansion at x=0

-1/5 - x + (50 x^3)/3 + (250 x^4)/3 + O(x^5)
(Taylor series)

Big‐O notation »

Derivative

d/dx(e^(10 x) (x - 1/5)) = e^(10 x) (10 x - 1)

Indefinite integral assuming all variables are real

integral(x - 1/5) e^(10 x) dx = e^(10 x) (x/10 - 3/100) + constant

Global minimum

$min{(x - 1/5) e^(10 x)} = -e/10 at x = 1/10$

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