How do you determine whether the function #y=ln x# is concave up or concave down and its intervals?

Question

How do you determine whether the function  #y=ln x# is concave up or concave down and its intervals?

Isabella Ackerman · Accepted Answer

The graph of #y = lnx# is concave down on #(0,oo)#.

The domain of #lnx# is #(0,oo)#. The second derivative is : #-1/x^2# which is always negative. So the graph of #y = lnx# is concave down on #(0,oo)#.

Christopher Allers · Answer

undefined To determine the concavity of the function $ y = \ln(x) $ and its intervals: 1. Find the second derivative $ y'' $ of $ y = \ln(x) $. 2. Set $ y'' $ equal to zero to find the points of inflection. 3. Test the intervals between these points of inflection with the second derivative test: - If $ y'' > 0 $ in an interval, the function is concave up on that interval. - If $ y'' < 0 $ in an interval, the function is concave down on that interval. For $ y = \ln(x) $: 1. First derivative $ y' = \frac{1}{x} $. 2. Second derivative $ y'' = -\frac{1}{x^2} $. Since $ y'' = -\frac{1}{x^2} $: - $ y'' > 0 $ for $ x > 0 $, indicating $ y = \ln(x) $ is concave up on $ (0, \infty) $. - $ y'' < 0 $ for $ x < 0 $, but the natural logarithm function is not defined for $ x \leq 0 $. Thus, there are no intervals where $ y = \ln(x) $ is concave down.