"How do you determine whether the function ln⁡(x2+10)..." was answered by our community

alahovom7ahc

Answered question

2022-04-15

How do you determine whether the function

\ln (x^{2} + 10)

is concave up or concave down and its intervals?

entreblogsmc2j

Beginner2022-04-16Added 10 answers

Step 1

f (x) = \ln (x^{2} + 10)

f^{'} (x) = \frac{2 x}{x^{2} + 10}

f x) = \frac{2 (10 - x^{2})}{{(x^{2} + 10)}^{2}}

f" is never undefined and is 0 at

\pm \sqrt{10}

Step 2
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

(- \sqrt{10}, \sqrt{10})

and concave down on

(- \infty, - \sqrt{10})

and on

(\sqrt{10}, \infty)

The points of inflection are:

(- \sqrt{10}, \ln 20) and (\sqrt{10}, \ln 20)

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