## 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.

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 x^{3}+3x^{2}+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.