Question

Analyzing a Graph In Exercise, analyze and sketch the graph of the function. Label any relative extrema, points of inflection, and asymptotes.

f(x) = 1/xe^x

Answer 1

Analyzed & Sketched

Explanation:

We are given the function:
`f(x) = (1)/(xe^x)`

So, we will find first derivative for finding the interval of increase and decrease.

`f'(x)=(-e^x-xe^x)/(x^2e^(2x)) =-(e^(-x)(x+1))/(x^2)`

it is increasing on (-∞, -1) and decreasing on (1,0) ∪ (0, ∞).

For concavity, we need to find the second derivative.

`f''(x)=((-e^x-e^x-xe^x)x^2e^(2x)+(e^x-xe^x)(2xe^(2x)+x^2e^(2x)))/(x^4e^(4x)) =(e^(-x)\left(x^2+2\left(x+1\right)\right))/(x^3)`

it is concave down on (-∞, 0) and concave up on (0, ∞).

x = 0 is vertical asymptote.

Sketch is given in the attachment.