What are the critical points of $f(x) =1/(xe^x)-e^(2x)/x$ ?

Question

1709 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-06-04T16:27:05

$"f"(x)=(1-e^(3x))/(xe^x)$

By using the series for the exponential function,

$"f"(x)=(1-(1+3x+(3x)^2/2!+...))/(x+x^2+x^3/2!+...)$

$"f"(x)=(3x+(3x^2)/2!+...))/(x+x^2+x^3/2!+...)$

$"f"(x)=(3+9x/2!+...)/(1+x+...)$

Then as $x \to 0$ , $"f"(x) \to 3$ .

What are the critical points of f(x) =1/(xe^x)-e^(2x)/xf(x) =1/(xe^x)-e^(2x)/x?