In other words, can we have something like a concave function where the second derivative is increasing over some range and then decreasing over another range?
Edit: What if the concave function is strictly increasing? (or decreasing?)
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Yes, $f(x)=-e^{x^{2}}$ is such a function.
answered Feb 7, 2020 at 5:21
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$\begingroup$ Thank you. Any chance there is an obvious example with an increasing, concave function, that you know of? (your example has the second derivative go from increasing to decreasing when the functions goes from increasing to decreasing. Other examples I have come up with -- such as $Ln[x^2]$ -- also have this property... $\endgroup$ Commented Feb 7, 2020 at 22:02