Concave and Convex Function

Concave and Convex Function

What is Concave Function?

Concave function is a function where the line segement between any two points of the function lies below or on the graph.[1]

Mathematically, as for concave function, the derivative of the left side of the max point is > 0, whereas the right side of the max point is < 0. There is one and only one value to make the derivative of the function equal to 0.

See the example below.

Concave Function

What is Convex Function?

Oppose to concave function, convex function is a function where the line segement between any two points of the function lies above or on the graph.[2]

Mathematically, as for concave function, the derivative of the left side of the max point is < 0, whereas the right side of the max point is > 0. There is one and only one value to make the derivative of the function equal to 0.

See the example below.

Convex Function

Neither Concave or Convex Function

As for other function which is neither concave or convex function, the number of the value to make the derivative of the function equal to 0 is multiple or none.

See the example below.

Neither

How to Find the Max/Min of Concave/Convex Function?

To find the max/min of concave/convex function, we need to make the value converge to the max/min.

So to converge the value, as for concave function, we should plus the derivative of the function, whereas we should substract the derivative of the function.

Let summary the attribute concave and convex function

Concave	Convex
Concave function is a function where the line segement between any two points of the function lies below or on the graph.	Oppose to concave function, convex function is a function where the line segement between any two points of the function lies above or on the graph.
Mathematically, as for concave function, the derivative of the left side of the max point is > 0, whereas the right side of the max point is < 0. There is one and only one value to make the derivative of the function equal to 0.	Mathematically, as for concave function, the derivative of the left side of the max point is < 0, whereas the right side of the max point is > 0. There is one and only one value to make the derivative of the function equal to 0.
Plus the derivative of the function	Substract the derivative of the function

See more detail below.

Converge

Reference

[1] Concave function - Wikipedia

[2] Convex function - Wikipedia