Concavity Convexity and Point of Inflexion

In mathematics, a function is a precise relationship between two sets (input set and output set). Every member of the output set has a unique relationship with one or more members of the input set. The letter “f” stands for the function. It’s feasible to have several different kinds of functions. They’re divided into groups according to their categories. The graph’s fundamental character is one of these categories. The functions can be classified into two types based on the graph’s nature:

• Function that is convex
• Function of concavity

A function’s concavity and convexity might occur once or multiple times in the same function. Alternatively, the location of an inflection point or point of inflection is referred to as an inflection point or point of inflection. The inflection point is where the function’s concavity changes. Depending on the direction of the arrow, the function changes from concave down to concave up, or vice versa. In other words, a tipping point is where the rate of change of slope shifts from increasing to decreasing, or vice versa. Those are not, by any stretch of the imagination, local maximum or minimum points. They are regarded to be permanent locations.

Concavity:

When a function’s curve bends, it usually takes on the shape of a concave cylinder. The word used to describe this phenomena is function concavity. In a graph function, there are two types of concavity that can be discovered.

• Concave upwards
• Downward concavity

Concave up or convex down describes an upward-opening curve or a curve that bends up into the shape of a cup, depending on which direction the curve opens or bends.

The fact that concave down or convex up curves bend down or resemble a cap in shape distinguishes them. In other words, if the tangent’s slope rises as a result of an increase in an independent variable, the tangent stays beneath the curve, and vice versa.

Convexity:

A convex function is a continuous function whose value at the midpoint of each interval in its domain does not exceed the arithmetic mean of the interval’s values.

When the set (x, y): x I, y f(x) is convex, we say that f(x) is convex on the interval I. When the set (x, y): x I, y f(x) is convex, we say that f is concave.

If the line segment connecting any two points on the graph of a real-valued function does not lie below the graph between the two points, the function is said to be convex in mathematics. A function is convex if its epigraph is a convex set, or vice versa.

Convex Functions’ Characteristics:

On the assumption that all functions are defined and continuous on the interval, we enumerate several features of convex functions.

If and are convex downhill (upward) functions, then any linear combination where and are positive real numbers is also convex downward (upward).

If one function is convex downward and the other is convex downward and non-decreasing, the composite function is convex downward as well.

The composite function is convex downward if one of the functions is convex upward and the other is convex downward and non-increasing.

On this interval, any local maximum of a convex upward function defined on the interval is also its global maximum.

Any convex descending function defined on the interval that has a local minimum is likewise its global minimum.

Finding concavity and convexity of a function:

For a function at first we have to find out the first derivative and then the second derivative, if the result is positive then it is convex. On the other side it is concave it the result is negative.

Inflection point:

The point of inflection, also known as the inflection point, is the point at which the function’s concavity changes. It denotes a change in function from concave down to concave up or vice versa. In other terms, an inflection point is the point at which the rate of change of slope changes from increasing to decreasing or vice versa. Those aren’t local peaks or minima, to be sure. They are points that are not moving.

How to find inflection point: 

At first we need to find out the second derivative of the function given. Then we have to put the second derivative is equal to zero to find out the potential inflection points.

Example: 

Given question : Y=x³ – 6x² + 12x

Answer : 

The first derivative of the given function is 3x² – 12x + 12

The second derivative of the given function  is 6x – 12 which is negative up to x=2 and positive after that.

So concave downward up to x = 2 and concave upward from x = 2

Point of inflexion of the given function is at x = 2.

Conclusion: 

Thus we know about concavity convexity and inflection point in this content. It is not only the mathematics only, but also significant to real life also. We can say that an inflection point is a turning point in the evolution of a firm, industry, sector, economy, or geopolitical scenario that might be deemed a turning point after which a major change, with either positive or negative consequences, is predicted.