Inflection Point Calculator

You want to evaluate the inflection point of a given function ? Try our inflection point calculator to find the inflection point of a graph where the concavity change.

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Introduction to Inflection Point Calculator:

Inflection point calculator is the best online source that helps you to evaluate the inflection point of a given function in a few seconds. It is used to find the given function inflection point on a graph where the concavity changes either in the upward direction or downward.

Inflection Point Calculator with Steps

It is a beneficial tool as it is used in various fields like mathematics,physics,engineering and economics where you need to analyse the change behaviour of a function curve on a graph.

What is the Inflection Point?

Inflection point is defined as a process in which you find inflection points where the given function f(x) changes its point of curvature on a graph in calculus. When the function changes its behaviour its concavity automatically changes.

If the function is in increasing order then the concave of a function is moving upward but if it is in decreasing order then concave changes its behaviour in downward direction.

How to Calculate Inflection Point?

For the Calculation of the inflection points of a function you need to use the differential method of finding the concavity where the function changes. Here’s a step-by-step guide about how to calculate the inflection points.

Step 1:

Determine the function f(x) to find the inflection point on the graph

Step 2:

Calculate the first derivative of a given function f′(x) as it gives you the slope of the function at any point.

Step 3:

Calculate the second derivative of the above differential function as f′′(x) to measure the rate of change of the first derivative to determine the concavity behaviour.

Step 4:

To find inflection point keep the second derivative function result equal to zero such as f′′(x)=0 to find point c around which concavity change

Step 5:

Follow the below condition under which you determine the nature of Inflection points,

If f′′(x)>0 at a point c then the function changes from concave up to concave down to represent the inflection point at a graph

If f′′(x)<0 at a point c the function changes from concave down to concave up, indicating the second inflection point at a graph.

By following these steps, you can easily find and understand the inflection points of different type of functions

Practical Example of Inflection Point:

The practical example of inflection point gives you an idea about how to evaluate the inflection point of a given function using a differential method.

Example: Find the inflection point of given function:

$$ f(x) \;=\; x^3 + 3x^2 + 1 $$

Solution:

The given function f(x) is,

$$ f(x) \;=\; x^3 + 3x^2 + 1 $$

Implement the differentiate the above function with respect to x,

$$ f’(x) \;=\; 3x^2 + 6x $$

Differentiate again with respect to x,

$$ Second\; derivative\; f’’(x) \;=\; 6x + 6 $$

To find the critical point put f``(x) = 0.

$$ Set\; f’’(x) \;=\; 0 $$

$$ 6x + 6 \;=\; 0 $$

$$ 6x \;=\; -6 $$

$$ x \;=\; -1 $$

So, x = -1 is a critical point.

To find the concavity change around x = -1. Consider x < -1

$$ f’’(-2) \;=\; 6(-2) + 6 \;=\; -12 + 6 \;=\; -6 $$

Consider x > -1 (e.g., x = 0)

$$ f’’(0) \;=\; 6(0) + 6 \;=\; 6 $$

For graphical representation of a given function x³+3x²+1 is given below,

PASTE THE GRAPH HERE!

How to Use Point of Inflection Calculator?

Points of inflection calculator has a user-friendly design that enables you to use it to easily calculate inflection point questions. Before adding the input of a function to find points of inflection, follow some simple steps. These steps are:

Enter the function in the input box to find the inflection point question with the solution.
Recheck your input function value before hitting the calculate button to start the calculation process in the inflection points calculator.
Click on the “Calculate” button to get the desired result of your given inflecton point questions with a solution.
If you want to try out our Inflection point calculator for the first time then you can use the load example to check the accuracy in solution.
Click on the “Recalculate” button to get a new page for solving inflection point problems to get solutions.

Final Result of Points Of Inflection Calculator:

Point of inflection calculator gives you the solution to a given function problem when you add the input value in it.It may contain as:

Result Step

When you click on the result option then it provides you with a solution for the inflection point question

Possible Step

When you click on the possible steps option it provides you with the solution of the inflection point problem in steps.

Plot Step

It provides you a solution in the form of a graph so that you get an understanding about the inflection point on a graph easily.

Advantages of Using Inflection Point Calculator:

The inflection points calculator has many advantages that you avail whenever you use it to calculate inflection points of a given function problem and get solutions without manual guidance. These advantages are:

It is a free tool so you can use it to find the inflection point problems with a solution without spending.
Point of inflection calculator is an adaptable tool as you can use it through electronic devices like laptops, computers, mobile, tablets, etc.
Our tool saves the time and effort that you consume in doing lengthy calculations of the inflection point of a given function f(x) in a few seconds.
It is a learning tool so you can use our tool for practice so that you get in-depth knowledge about inflection points and concavity.
It provides you solutions in a complete process in a step-by-step method for a better understanding of inflection point questions.
It is a reliable tool that provides you with accurate solutions according to your input whenever you use the points of inflection calculator to get a result.

Related References

What is inflection point?
Give some examples of inflection point:
How to evaluate inflection point?
Explain inflection point with the help of examples.

Frequently Ask Questions

How to Find Points of Inflection?

Step 1: Find the second derivative:

Given a function f(x), the points of inflection occur where the concavity of the function changes. To determine this, calculate the second derivative, denoted as f′′(x).

Step 2: Set the second derivative equal to zero:

Solve the equation f′′(x) = 0 to find potential inflection points. These are the candidate points where the concavity might change.

Step 3: Test the concavity around these points:

To confirm that a point is indeed an inflection point, check the concavity by examining the sign of the second derivative f′′(x) on either side of each candidate point:

If the sign of f′′(x) changes from positive to negative or negative to positive as you pass through the point, then it is a point of inflection.
If the sign does not change, then the point is not an inflection point.

How to Find Points of Inflection on a Graph

Observe the Concavity:

Concave Up: If the graph curves upward like a cup (i.e., it looks like a "U"), the function is concave up. This happens when the second derivative f′′(x) is positive.
Concave Down: If the graph curves downward like a frown (i.e., it looks like an "n"), the function is concave down. This happens when the second derivative f′′(x) is negative.

Locate Where the Concavity Changes:

A point of inflection occurs when the graph transitions from concave up to concave down, or vice versa. It is typically where the graph flattens momentarily before changing direction.
Look for spots where the curve seems to "switch" its direction of bending.

Identify Key Characteristics:

The point of inflection may not always be a local maximum or minimum.
At the inflection point, the slope of the graph might not be zero; the tangent line could still have a non-zero slope.

Test Nearby Intervals:

To confirm an inflection point, compare the concavity immediately to the left and right of the suspected inflection point:

If the graph is concave down on one side and concave up on the other, you've likely found an inflection point.

Example on a Graph:

Imagine a graph with a cubic-like curve:

From x = −2 to x = 0, the graph curves downward (concave down).
From x = 0 to x = 2, the graph curves upward (concave up).

In this case, x = 0 is a potential point of inflection because the concavity changes from down to up.

How to Find Horizontal Points of Inflection

A horizontal point of inflection occurs when the curve flattens (the slope is zero) and changes its concavity, meaning the direction of its bending shifts.

In simple terms, it's a point where the graph looks flat for a moment but switches from curving upwards to downwards (or vice versa). While the curve doesn’t reach a peak or trough at this point, the inflection indicates a significant shift in the shape of the graph.

To identify these points, both the first derivative (which gives the slope) and the second derivative (which shows the concavity) of the function play a role.

The slope must be zero at the point, and the concavity should change on either side of it. This makes horizontal inflection points distinct, as the graph’s curvature alters while remaining flat at that specific location.

How to find Inflection Points of a Polynomial Function

Find the Second Derivative:

Start by finding the first derivative f′(x)f of the polynomial function to get the slope. Then, take the derivative of f′(x) to get the second derivative f′′(x). The second derivative gives you information about the concavity of the graph.

Set the Second Derivative Equal to Zero:

Solve the equation f′′(x)=0. The solutions will give you the critical points where the concavity may change. These points are potential candidates for inflection points.

Check for a Sign Change in Concavity:

To confirm whether each candidate point is truly an inflection point, you need to check if the concavity changes at that point. Pick values of x slightly to the left and right of each candidate point and plug them into the second derivative f′′(x):

If f′′(x) changes from positive to negative (concave up to concave down) or from negative to positive (concave down to concave up), then the point is an inflection point.

Example:

Consider the polynomial function $$ f(x) \;=\; x^3 - 6x^2 + 9x + 1 $$

First derivative:
$$ f'(x) \;=\; 3x^2 - 12x + 9 $$
Second derivative:
$$ f′′(x) \;=\; 6x − 12 $$
Set f′′(x) = 0 and solve:
$$ 6x − 12 \;=\; 0 $$
$$ x \;=\; 2 $$
This is a candidate point for an inflection point.
Test the concavity on either side of x = 2x = 2x = 2:

At x=1, f′′(1) = 6(1) − 12 = −6 (concave down).
At x=3, f′′(3) = 6(3) − 12 = 6(concave up).

Since the concavity changes from down to up, x=2x = 2x=2 is an inflection point.

Amanda Jepson

Last Updated February 21, 2024