Question

How do you determine whether the function `f(x) = ln(x^2 + 7)` is concave up or concave down and its intervals?

Answer 1

Concave down over `(-oo ; -sqrt 7) uu (sqrt 7 ; oo)`
Concave up over `(-sqrt7 ; sqrt7 )`

Explanation:

We first find the second derivative of the function using normal rules of differentiation.

We then find the points where this second derivative s either zero or undefined. These are the inflection points where the concavity changes.

We then investigate the sign of the second derivative inbetween and around the inflection points to decide on which intervals are concave up and/or down.

The results are given in the attached sketch and the graph of the function is also given for completeness sake to observe the concavity in each interval.

graph{ln(x^2+7) [-10.21, 9.79, -2.76, 7.24]}