Inflection Point

What is an Inflection Point?

In Mathematics, a function is stated as the relationship between two sets i.e. input set and the output set. Each element of the output is distinctively related to the one or more elements of the input set. The function is denoted by (f). There are distinct types of functions. They are classified according to their class. One such class is the nature of graphs. On the basis of the nature of the graph, the function is divided into two types such as :

  1. Convex Function

  2. Concave Function

Both the concavity and convexity appears in a function once or more than once. The point where the function is neither considered as convex or concave is known as the inflection point. In this article, we will study what is an inflection point, inflection point graph, inflection point definition, how to find inflection points on graphs, inflection point examples etc.


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Inflection Point Definition

The inflection point definition stated that it is a point where the concavity of a function does not vary. It implies that function varies from concave up to concave down or vice versa. In other words, it states that inflection point is the point in which the rate of slope changes in increasing to decreasing order or vice versa. These points are generally not local maxima or minima but stationary points.

 

Concavity Function

A concave shape is formed when the curve of a function bends down. It is known as the concavity of a function. We can see two types of concavity in the inflection point graphs. These two types of concavity found in inflection point graph are

  1. Concave up

  2. Concave down

Concave Up - A curve is said to be concave up if it opens in an upward direction or bends up to mold a shape like a cup. 


Concave Down - A curve is said to be concave down if it opens in the downward direction or bends down to mold a shape like a cup. In other words, tangent lies beneath the curve if the slope of the tangent increases by the increases in an independent variable.


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If f(z) is differentiable, the f(z) is represented as 


Concave up a point  z = k, ifff f “ (z) > 0 at k


Concave down at a point x = k, ifff f “ (z) < 0 at k


Inflection Point Graph 

Here, you can see the inflation point graph with its two types of concavity i.e. concave up and concave down.


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The point of inflection represents the slope of a graph of a function in which the specific point is zero. The above inflection point graph shows that the function has an inflection point.


It is important to note that in a single curve or within the given interval of a function, there can be one or more than one point of inflection.


How to Find the Inflection Point on a Graph?

An inflection point is represented as the point on the curve in which concavity varies i.e. the sign of the curvature varies. And we have read above, that concave up point represents if f’’ > 0 and concave down point represents if f“<0. If the function changes from positive to negative or negative to positive at a particular point x = c, then the point is considered as a point of inflection on a graph.


Solved Examples

1. Find the point of inflection and intervals of concavity up and down of.


f(x)= 2x³ - 12x² + 4x - 27


Solution: The second derivative in the above equation is f’’(x) = 12x - 4 = 0


Hence, solving 12x - 24 = 0


There is only just a single inflection point and i.e. 2. Now, Select the auxiliary points  t₀ = 0 to the left side of the point of inflection and t1 = 3 to the right side of the point of inflection.


Then f’’(x) = -24 < 0, so on (- ∞,2) the curve is concave downward and f’’(3) = 12 > 0, so on ( 2,∞) the curve is concave upward.


2. Find the inflection point of the function given below


f(x) =  x³ - 2.3x² - 1.4x


Solution:

Differentiate f(x) with respect to x, we get:


f’(x) = 3x² - 2.3(2) x- 1.4


f’(x) = 3x² - 4.6x - 1.4


Differentiate f(x) again, we get


f’’(x) = 3(2) x - 4.6 - 0


f’’(x) = 6x - 4.6


Substitute f’’(x) = 0, we get:


6x - 4.6 = 0


x= 4.6/6


x= 23/30


Hence, the x- coordinate of the inflection point of the given function is x = 23/30.


Quiz Time

1. Examine the function f(x) given below. At x=0 is this function concave up, concave down or an inflection point.

F (x) = ex - x

  1. Concave up

  2. Concave down

  3. An inflection point

  4. This function is not differentiable


2. Examine the function f(x) given below. At x=0 is this function concave up, concave down or an inflection point.


F (x) = ½ cos(πx)ex

  1. Concave up

  2. Concave down

  3. An inflection point

  4. This function is not differentiable

FAQ (Frequently Asked Questions)

1. What are the Steps to Find the Inflection of a Point?

As we know, the inflection point is a point where the curve varies from concave upward to concave downward or concave downward to concave upward.


Here, we will learn the steps to find the inflection of a point.

  1. Differentiate the function f(z), to get f(z)

  2. Solve the equation f(z) = 0 to receive the values of z at minima or maxima or point of inflection

  3. For each z values:

  • Find out the values of f(z) for values a smaller and a little larger than z value.

  • Determine whether you have a maximum or minimum point of inflection

  • Calculate the values of a function at the z value for the point of inflection.

2. What is the Use of the Second Derivative in Determining the Point of Inflection?

In calculus, the derivative is useful in several ways. The most widely used derivative is to find the slope of a line tangent to a curve at a given point. The derivation is also used to find the inflection point of the graph of a function.


If the graph y = f(x) has an inflection point at x = z, then the second derivative of f evaluated at z is 0. We can represent this mathematically as f’’ (z) = 0. If the second derivative of a function is 0 at a point, this does not mean that point of inflection is determined. However, we can check for potential inflection point by looking at which point the second derivative will be zero 


We will use this method to determine the location of the inflection points of the normal distribution.