How do you determine whether the function #ln(x^2+10)# is concave up or concave down and its intervals?

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Answer 1 · 2015-08-31T14:46:44

Investigate the sign of the second derivative.

#f(x) = ln(x^2+10)#

#f'(x) = (2x)/(x^2+10)#

#f''(x) = (2(10-x^2))/(x^2+10)^2#

#f''# is never undefined and is #0# at #+-sqrt10#

The sign of #f''# is the same as the sign of #10-x^2#.

It is positive near #0# and negative far from #0#.

So the graph of #f# is concave up on #(-sqrt10, sqrt10)#

and concave down on #(-oo, -sqrt10)# and on #(sqrt10, oo)#.

The points of inflection are:

#(-sqrt10, ln20)# and #(sqrt10, ln20)#