How do you determine whether the function $ln (x^{2} + 10)$ $ln(x^2+10)$ 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.

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

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

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

$f''$ 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 $(sqrt10, ln20)$

How do you determine whether the function ln(x^2+10)ln(x2+10)ln(x^2+10) is concave up or concave down and its intervals?