Interval of Concavity for

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  • #1
hannahSUU
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Find the interval where the function is concave up and down.

The answers I got are:
Concave up: (ln(2), infinity)
Concave down: (0, ln(2))

and for the life of me I cannot figure out why they aren't correct.

any help? (heart)
 
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  • #2
hannahSUU said:


Find the interval where the function is concave up and down.

The answers I got are:
Concave up: (ln(2), infinity)
Concave down: (0, ln(2))

and for the life of me I cannot figure out why they aren't correct.

any help? (heart)

Can you show your working in computing the second derivative?
 
  • #3
As a follow-up, we are given:



The first thing I would observe is that the domain of this function is .

Using the product rule, we find:



Using the product rule again, we find:



As , we need only look at the critical value that comes from:



Using test values on either side of this critical value, such as or , we see that we have:

on and thus is concave down on this interval.

on and thus is concave up on this interval.

Thus, we have a point of inflection at:



Here is a plot of the given function with the point of inflection shown:

View attachment 7538
 

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