In differential calculus, an inflection point is a point on a curve where the curvature changes its sign (from positive to negative or vice versa). It is used in various subjects, including engineering, economics, and statistics, to bring about fundamental changes within data. If you need to find an inflection point in a curve, go to Step 1.
Steps
Method 1 of 3: Understanding the Inflection Points
Step 1. Understanding concave functions
To understand inflection points, you need to distinguish concave from convex functions. A concave function is a function in which, taken any line connecting two points of its graph, never lies above the graph.
Step 2. Understanding convex functions
A convex function is essentially the opposite of a concave function: it is a function in which any line connecting two points on its graph never lies below the graph.
Step 3. Understanding the root of a function
A root of a function is the point at which the function equals zero.
If you were to graph a function, the roots would be the points where the function intersects the x axis
Method 2 of 3: Find the Derivatives of a Function
Step 1. Find the first derivative of the function
Before you can find the inflection points, you will need to find the derivatives of your function. The derivative of a basis function can be found in any analysis text; you have to learn them before you can move on to more complex tasks. The first derivatives are denoted by f ′ (x). For polynomial expressions of the form axp + bx(p − 1) + cx + d, the first derivative is apx(p − 1) + b (p - 1) x(p − 2) + c.
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For example, suppose you need to find the inflection point of the function f (x) = x3 + 2x − 1. Calculate the first derivative of the function as follows:
f ′ (x) = (x3 + 2x - 1) ′ = (x3) ′ + (2x) ′ - (1) ′ = 3x2 + 2 + 0 = 3x2 + 2
Step 2. Find the second derivative of the function
The second derivative is the derivative of the first derivative of the function, denoted by f ′ ′ (x).
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In the example above, the second derivative will look like this:
f ′ ′ (x) = (3x2 + 2) ′ = 2 × 3 × x + 0 = 6x
Step 3. Equal the second derivative to zero
Match your second derivative to zero and find the solutions. Your answer will be a possible inflection point.
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In the example above, your calculation will look like this:
f ′ ′ (x) = 0
6x = 0
x = 0
Step 4. Find the third derivative of the function
To understand if your solution is indeed an inflection point, find the third derivative, which is the derivative of the second derivative of the function, denoted by f ′ ′ ′ (x).
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In the example above, your calculation will look like this:
f ′ ′ ′ (x) = (6x) ′ = 6
Method 3 of 3: Find the inflection point
Step 1. Evaluate the third derivative
The standard rule for calculating a possible inflection point is as follows: "If the third derivative is not equal to 0, then f ′ ′ ′ (x) ≠ 0, the possible inflection point is actually an inflection point." Check your third derivative. If it is not equal to 0 at the point, it is a real inflection.
In the example above, your computed third derivative is 6, not 0. Therefore, it is a real inflection point
Step 2. Find the inflection point
The coordinate of the inflection point is denoted as (x, f (x)), where x is the value of the variable x at the inflection point and f (x) is the value of the function at the inflection point.
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In the example above, remember that when you calculate the second derivative, you find that x = 0. So, you need to find f (0) to determine the coordinates. Your calculation will look like this:
f (0) = 03 + 2 × 0−1 = −1.
Step 3. Write down the coordinates
The coordinates of your inflection point are the x value and the value calculated above.
