Question

Consider the function g(x) = (x-e)^3e^-(x-e). Find all critical points and points of inflection (x, g(x)) of the function g.

Answer 1

The answer is "
`cirtical\ points \ (x,g(x))\equiv (e,0),(e+3,(27)/(e^3))`"

Explanation:

Given:

`g(x) = (x-e)^3e^(-(x-e))`

Find critical points:

`g(x) = (x-e)^3e^((e-x))`

differentiate the value with respect of x:

`\to g'(x)= (x-e)^3 (d)/(dx)e^(e-r) +e^(e-r) (d)/(dx)(x-e)^3=(x-e)^2 e^((e-x)) [-x+e+3]`

critical points
`g'(x)=0`

`\to (x-e)^2 e^((e-x)) [e+3-x]=0\\\\\to e^((e-x))\\eq 0 \\\\\to (x-e)^2=0\\\\ \to [e+3-x]=0\\\\\to x=e\\\\\to x=e+3\\\\\to x= e,e+3`

So,

The critical points of
`(x,g(x))\equiv (e,0),(e+3,(27)/(e^3))`