Question

Suppose that f(x)=ln(5+x^2). Use interval notation to indicate where f(x) is concave up and concave down.

Answer 1

2nd derivative determines concavity
f''=0 inflection point
f'' >0 concave up
f'' <0 concave down

\[f'(x) = \frac{2x}{x^{2}+5}\]
\[f''(x) = \frac{10 -2x^{2}}{(x^{2}+5)^{2}}\]
Inflection points:
\[x = \pm \sqrt{5}\]
\[f''(0) = \frac{10}{25} > 0 \]
thus from (-sqrt5, sqrt5) f(x) is concave up
from (-inf,-sqrt5] and [sqrt5,inf) f(x) is concave down

Answer 2

Thanks :]

Answer 3

your welcome

Answer 4

do you get how i got the derivatives?

Answer 5

yeah, second derivative of f(x). I was ignoring the inflection points :/

Answer 6

yeah inflection points tell you when concavity changes