Inflection Point

In Mathematics, a function is a special relationship between two sets (input set and output set). Every member of the output set is uniquely related to one or more members of the input set. The function is represented by “f”. There are different types of functions. They are classified according to the categories. One such category is the nature of the graph. Depending upon the nature of the graph, the functions can be divided into two types namely

• Convex Function
• Concave Function

Both the concavity and convexity can occur in a function once or more than once. The point where the function is neither concave nor convex is known as inflection point or the point of inflection. In this article, the concept and meaning of inflection point, how to determine the inflection point graphically are explained in detail.

Inflection Point Definition

The point of inflection or inflection point is a point in which the concavity of the function changes. It means that the function changes from concave down to concave up or vice versa. In other words, the point in which the rate of change of slope from increasing to decreasing manner or vice versa is known as an inflection point. Those points are certainly not local maxima or minima. They are stationary points.

Concavity Function

Generally, when the curve of a function bends, it forms a concave shape. It is known as the concavity of a function. In graph function, two types of concavity can be found.

• Concave up
• Concave down

Concave Up – If a curve opens in an upward direction or it bends up to make a shape like a cup, it is said to be concave up or convex down.

Concave Down – If a curve bends down or resembles a cap, it is known as concave down or convex up. In other words, the tangent lies underneath the curve if the slope of the tangent increases by the increase in an independent variable.

Inflection Point Calculus

If f(x) is a differentiable function, then f(x) is said to be:

• Concave up a point x = a, iff f “(x) > 0 at a
• Concave down at a point x = a, iff f “(x) < 0 at a

Here, f “(x) is the second order derivative of the function f(x).

Inflection Point Graph

The point of inflection defines the slope of a graph of a function in which the particular point is zero. The following graph shows the function has an inflection point.

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

How to Find the Inflection Point on a Graph?

An inflection point is defined as a point on the curve in which the concavity changes. (i.e) sign of the curvature changes. We know that if f ” > 0, then the function is concave up and if f ” < 0, then the function is concave down. If the function changes from positive to negative, or from negative to positive, at a specific point x = c, then that point is known as the point of inflection on a graph.

Inflection Point of a Function

We can identify the inflection point of a function based on the sign of the second derivative of the given function.  Also, by considering the value of the first-order derivative of the function, the point inflection can be categorized into two types, as given below.

If f'(x) is equal to zero, then the point is a stationary point of inflection.

If f'(x) is not equal to zero, then the point is a non-stationary point of inflection.

Click here to get the inflection point calculator.

Inflection Point Examples

Refer to the following problem to understand the concept of an inflection point.

Example:

Determine the inflection point for the given function f(x) = x⁴ – 24x²+11

Solution:

Given function: f(x) = x⁴ – 24x²+11

The first derivative of the function is

f’(x) = 4x³ – 48x

The second derivative of the function is

f”(x) = 12x² – 48

Set f”(x) = 0,

12x² – 48 = 0

Divide by 12 on both sides, we get

x² – 4 = 0

x² = 4

Therefore, x = ± 2

To check or x = 2, substitute x= 1 and 3 in f”(x)

So, f”(1) = 12(1)² – 48 = -36 (negative)

f”(3) = 12(3)² – 48 = 276 (positive)

To check for x = -2, substitute x= 0 and -3 in f”(x)

So, f”(0) = 12(0)² – 48 = -48 (negative)

f”(3) = 12(3)² – 48 = 276 (positive)

Hence, proved

Now, substitute x = ± 2 in f”(x)

Therefore, it becomes

f”(2) = 12(2)² – 48 = -69

f”(-2) = 12(-2)² – 48 = -69

Therefore, the inflection points are (2, -69), and (-2, -69).

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