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

2 Answers
Sep 16, 2015

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

Sep 16, 2015

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.)