Question

How do you determine whether the function `h '(x) = ((x^2) - 2) / x` is concave up or concave down and its intervals?

Answer 1

concave up

Explanation:

find the derrivaive to the given function
it will be `1+2/x^2`
for every integer value of x the value decreases
slope of the function `f(x)` i e`f'(x`) here the `f` is `h'`
so as value of `h"` is decreasing
the function `h'(x)` is concave up

Answer 2

The function `f(x) = (x^2-2)/x` is concave up on `(-oo,0)` and concave down on `(0,oo)`. There is no inflection point.

Explanation:

Investigate the sign of the second derivative.

`f(x) = (x^2-2)/x` may be easier to differentiate if we write it as

`f(x) = x-2/x`

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

`f''(x) = -2/x^3`

So
`f''(x)` is positive and the graph of `f` is concave up on `(-oo,0)`
and
`f''(x)` is negative and the graph of `f` is concave down on `(0,oo)`

Because `f` is not defined at `0`, there is no point of the graph at which the concavity changes. (I.e. there is no inflection point.)