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Inflection Point Calculator

Write down any function and the free inflection point calculator will instantly calculate concavity solutions and find inflection points for it, with the steps shown.


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Use this free handy Inflection point calculator to find points of inflection and concavity intervals of the given equation. Apart from this, calculating the substitutes is a complex task so by using this point of inflection calculator you can find the roots and type of slope of a given function.

Here’s, you can explore when concave up and down and how to find inflection points with derivatives.

What is an Inflection Point?

In Calculus, an inflection point is a point on the curve where the concavity of function changes its direction and curvature changes the sign. In other words, the point on the graph where the second derivative is undefined or zero and change the sign.

Similarly, The second derivative f’’ (x) is greater than zero, the direction of concave upwards, and when f’’ (x) is less than 0, then f(x) concave downwards.

Inflection Point Calculator

In order to find the inflection point of the function Follow these steps.

Take a quadratic equation to compute the first derivative of function f'(x).

Now perform the second derivation of f(x) i.e f''(x) as well as solve 3rd derivative of the function.

Third derivation of f'''(x) should not be equal to zero and make f''(x) = 0 to find the value of variable.

Substitutes of x value in 3rd derivation of function to know the minima and maxima of the function.

Replace the "x" value in the given function to get the "y" value.

Then, the inflection point will be the x value, obtain value from a function.


Find the inflection points for the function \(f(x) = -2x^4 + 4x^2\)?


Given function is = \(-2x^4 + 4x^2\)

$$f^(x) = -8x^3 + 8x$$

$$f^{''}(x) = -24x^2 + 8$$

$$F^{'''}(x) = -48x$$

By taking second derivative

$$f^{''}(x) = 0$$

$$-24x^2 + 8 = 0$$

$$24x^2 = 8$$

Divide by 8 both sides

$$3x^2 = 1$$

$$x^2 = \frac{1}{3}$$

$$x = ± \frac{\sqrt{3}}{3}$$

Substitute of \(x = ± 1\) in function \(f^{'''}(x)\).

Condition for an Inflection Point ( Derivative Test):

When x_0 is the point of inflection of function f(x) and this function has second derivative f’’ (x) from the vicinity of x_0, that continuous at point of x_0 itself, then it states

$$f^{''}(x_0) = 0$$

However, we can find necessary conditions for inflection points of second derivative f’’ (x) test with inflection point calculator and get step-by-step calculations.

Moreover, an Online Derivative Calculator helps to find the derivation of the function with respect to a given variable and shows complete differentiation.

First Sufficient Condition for Inflection Point:

If the function is differentiable and continuous at a point x_0, has a second derivative in some deleted neighborhood of the point x_0, and if the second derivative changes slope direction when passing through the point x_0, then x_0 is a point of inflection of the function.

Second Sufficient Condition for an Inflection Point:

The x_0 is the inflection point of the function f(x) when the second derivation is equal to zero but the third derivative f’’’ (x_0) is not equal to zero.

$$F'' (x_0) = 0$$

$$F''' (x_0) ≠ 0$$

How to Find Concavity?

A graph has concave upward at a point when the tangent line of a function changes and point lies below the graph according to neighborhood points and concave downward at that point when the line lies above the graph in the vicinity of the point. So, the concave up and down calculator finds when the tangent line goes up or down, then we can find inflection point by using these values.

Hence, the graph of derivative y = f’ (x) increased when the function y = f(x) is concave upward as well as when the derivative y = f’ (x) decreased the function is concave downward and the graph derivative y = f’(x) has minima or maxima when function y = f(x) has an inflection point.

Furthermore, an Online Slope Calculator allows you to find the slope or gradient between two points in the Cartesian coordinate plane.

How Inflection Point Calculator works?

To find inflection points with the help of point of inflection calculator you need to follow these steps:


  • First, enter a quadratic equation to determine the point of inflection, and the calculator displays an equation that you put in the given field.
  • Now, press the calculate button.


When you enter an equation the points of the inflection calculator gives the following results:

  • It shows inflection points according to entered values also displays the points when concave up and down with its substitutes.
  • Moreover, it tells the tangent line rise or fall and shows the first, the second, and third derivative of the function f(x) with complete calculation.


How we Get Maxima, Minima, and Inflections Points with Derivatives?

The relative extremes can be the points that make the first derivative of the function which is equal to zero:

F’(x_0) = 0

These points will be a maximum, a minimum, and an inflection point so, they must meet the second condition.

How do know Maximums, Minimums, and Inflection Points?

Once we get the points for which the first derivative f’(x) of the function is equal to zero, for each point then the inflection point calculator checks the value of the second derivative at that point is greater than zero, then that point is minimum and if the second derivative at that point is f’’(x)<0, then that point is maximum.

What is the Stationary and Non-Stationary Point Inflection?

  • When f’(x) is equal to zero, the point is stationary of inflection.
  • The point is the non-stationary point of inflection when f’(x) is not equal to zero.

Final Point:

An inflection point calculator is specifically created by calculator-online to provide the best understanding of inflection points and their derivatives, slope type, concave downward and upward with complete calculations. Undoubtedly, you can get these calculations manually with the help of a graph but it increases the uncertainty, so you have to choose this online concavity calculator to get 100% accurate values.


From the source of Wikipedia: A necessary but not sufficient condition, Inflection points sufficient conditions, Categorization of points of inflection.

From the source of Dummies: Functions with discontinuities, Analyzing inflection points graphically.

From the source of Khan Academy: Inflection points algebraically, Inflection Points, Concave Up, Concave Down, Points of Inflection.      

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