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

Sep 16, 2015

concave up

Explanation:

find the derrivaive to the given function
it will be $1 + \frac{2}{x} ^ 2$
for every integer value of x the value decreases
slope of the function $f \left(x\right)$ i ef'(x) here the $f$ is $h '$
so as value of h" is decreasing
the function $h ' \left(x\right)$ is concave up

Sep 16, 2015

The function $f \left(x\right) = \frac{{x}^{2} - 2}{x}$ is concave up on $\left(- \infty , 0\right)$ and concave down on $\left(0 , \infty\right)$. There is no inflection point.

Explanation:

Investigate the sign of the second derivative.

$f \left(x\right) = \frac{{x}^{2} - 2}{x}$ may be easier to differentiate if we write it as

$f \left(x\right) = x - \frac{2}{x}$

$f ' \left(x\right) = 1 + \frac{2}{x} ^ 2$ and

$f ' ' \left(x\right) = - \frac{2}{x} ^ 3$

So
$f ' ' \left(x\right)$ is positive and the graph of $f$ is concave up on $\left(- \infty , 0\right)$
and
$f ' ' \left(x\right)$ is negative and the graph of $f$ is concave down on $\left(0 , \infty\right)$

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.)