What is a concave function and how is it determined?

[kaliprasad]

I was reading some where that $\sqrt{x}$ is concave function what does it mean.

 

[I like Serena]

Re: meaing of concave function

I was reading some where that $\sqrt{x}$ is concave function what does it mean.

Hi kaliprasad,

From wiki, a real-valued function defined on an interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph.
A concave function is the negative of a convex function.

Since the line segment between any 2 points on the graph of $\sqrt x$ lies strictly below the graph, it is strictly concave.

 

[MarkFL]

Re: meaing of concave function

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 $f''<0$.

 

[HallsofIvy]

Re: meaing of concave function

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 $f''<0$.

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.

 

[kaliprasad]

Thanks to all of you for the same. The property mentioned by MARKFL helps in chcking if the function is concave