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

  • Thread starter Thread starter kaliprasad
  • Start date Start date
  • Tags Tags
    Concave Function
AI Thread Summary
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.
kaliprasad
Gold Member
MHB
Messages
1,333
Reaction score
0
I was reading some where that $\sqrt{x}$ is concave function what does it mean.
 
Mathematics news on Phys.org
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
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top