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

Question

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

Samantha Adkison · Accepted Answer

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

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]}

Axel Alvarez · Answer

undefined To determine the concavity of the function $ f(x) = \ln(x^2 + 7) $ and its intervals, we need to find the second derivative and analyze its sign. First, find the first derivative of $ f(x) $ and then find the second derivative. Once you have the second derivative, analyze its sign to determine concavity.

1. Find the first derivative:
$$ f'(x) = \frac{d}{dx}\left(\ln(x^2 + 7)\right) $$

2. Find the second derivative:
$$ f''(x) = \frac{d^2}{dx^2}\left(\ln(x^2 + 7)\right) $$

3. Analyze the sign of $ f''(x) $ to determine concavity:
   - If $ f''(x) > 0 $ on an interval, the function is concave up on that interval.
   - If $ f''(x) < 0 $ on an interval, the function is concave down on that interval.

4. Determine the intervals where the function is concave up or concave down by finding where the second derivative changes sign, or by testing intervals within the domain of the function.