Question

If `( x+2) / x`, what are the points of inflection, concavity and critical points?

Answer 1

Please see the explanation below

Explanation:

Let `f(x)=(x+2)/x`

The domain of `f(x)` is `x in RR-{0}`

The critical points are when `f'(x)=0`

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

`f'(x)!=0`

There are no critical points

The points of inflection are when `f''(x)=0`

`f''(x)=4/x^3`

`f''(x)!=0`

There are no points of inflections

The concavity will depend on the sign of `f''(x)`

Let's build a sign chart

`color(white)(aaaa)` `"Interval"` `color(white)(aaaa)` `(-oo,0)` `color(white)(aaaa)` `(0, +oo)`

`color(white)(aaaa)` `"Sign " f''(x)"` `color(white)(aaaa)` `-` `color(white)(aaaaaaa)` `+`

`color(white)(aaaa)` `"Concavity"` `color(white)(aaa)` `concave` `color(white)(aaaa)` `convex`

graph{(x+2)/x [-10, 10, -5, 5]}