MHB What is a concave function and how is it determined?

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    Concave Function
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A concave function, such as $\sqrt{x}$, is defined as one where the line segment between any two points on its graph lies strictly below the graph itself. This characteristic indicates that the function is concave downward, which is confirmed by its negative second derivative, $f''(x) = -\frac{1}{4x^{\frac{3}{2}}}$, for $x > 0$. The discussion emphasizes that while a negative second derivative is a common way to identify concavity, the formal definition involves the convexity of the set of points above the graph. Understanding these properties helps in determining whether a function is concave or convex. The insights shared clarify the mathematical definitions and properties of concave functions.
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I was reading some where that $\sqrt{x}$ is concave function what does it mean.
 
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Re: meaing of concave function

kaliprasad said:
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.
 
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 $f''<0$.
 
Re: meaing of concave function

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 $f''<0$.
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.
 
Thanks to all of you for the same. The property mentioned by MARKFL helps in chcking if the function is concave
 
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