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

Question

Isabella Ackerman · Accepted Answer

Investigate the sign of the second derivative.

#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)#

Paisley Johnson · Answer

undefined To determine the concavity of the function $ \ln(x^2 + 10) $ and its intervals, we need to find the second derivative of the function and analyze its sign.

1. Find the first derivative:
$$ f'(x) = \frac{d}{dx} \ln(x^2 + 10) = \frac{1}{x^2 + 10} \cdot 2x = \frac{2x}{x^2 + 10} $$

2. Find the second derivative:
$$ f''(x) = \frac{d}{dx} \left( \frac{2x}{x^2 + 10} \right) $$

To find $ f''(x) $, you can use the quotient rule or simplify the expression first.

After finding $ f''(x) $, determine its sign to identify the concavity of the function and its intervals. If $ f''(x) > 0 $, the function is concave up in that interval, and if $ f''(x) < 0 $, the function is concave down in that interval.