The discussion revolves around the concept of concave functions, specifically examining the function $\sqrt{x}$ and how its concavity is determined. Participants explore definitions, properties, and methods of identifying concavity, including the relationship with second derivatives.
Participants present multiple viewpoints on the definition and identification of concave functions, with some agreement on the use of second derivatives, but no consensus on a singular definition or approach.
There are nuances in the definitions of convex and concave functions that are discussed, including the relationship between geometric interpretations and calculus-based criteria. The discussion does not resolve these nuances.
kaliprasad said:I was reading some where that $\sqrt{x}$ is concave function what does it mean.
That is, of course true, and often the easiest way to use "convex function", but is not the definition of "convex function". A set is "convex" if and only if, given any two points, A and B, in that set the line segment between A and B is also in the set. A function, f, is said to be "convex" ("convex upward" is typically implied by "convex" alone) if and only if the set of all points above the graph of y= f(x) is a convex set. The function is "convex downward" if the set of all points below the graph of y= f(x) is a convex set.MarkFL said:I had always thought of a concave function, or concave down, over some interval, as having a negative second derivative on that interval. If we have:
$$f(x)=\sqrt{x}$$
then we find:
$$f''(x)=-\frac{1}{4x^{\frac{3}{2}}}$$
Hence, we see that on the interval $(0,\infty)$ we have $f''<0$.