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Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com PDF Download

Inflection Points 

An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa)

So what is concave upward / downward ?
Concave upward is when the slope increases:
Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com
Concave upward is when the slope decreases:
Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com
 Here are some more examples:

Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Finding where ...

So our task is to find where a curve goes from concave upward to concave downward (or vice versa).

Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Calculus

Derivatives help us!

The derivative of a function gives the slope.

The second derivative tells us if the slope increases or decreases.

  • When the second derivative is positive, the function is concave upward.
  • When the second derivative is negative, the function is concave downward.

And the inflection point is where it goes from concave upward to concave downward (or vice versa).
Example: y = 5x3 + 2x2 − 3x

Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Let's work out the second derivative:

  • The derivative is y' = 15x2 + 4x − 3
  • The second derivative is y'' = 30x + 4

And 30x + 4 is negative up to x = −4/30 = −2/15, positive from there onwards. So:

f(x) is concave downward up to x = −2/15

f(x) is concave upward from x = −2/15 on

And the inflection point is at x = −2/15
 

A Quick Refresher on Derivatives

In the previous example we took this:

y = 5x3 + 2x2 − 3x

and came up with this derivative:

y' = 15x2 + 4x − 3

There are rules you can follow to find derivatives, and we used the "Power Rule":

  • x3 has a slope of 3x2, so 5x3 has a slope of 5(3x2) = 15x2
  • x2 has a slope of 2x, so 2x2 has a slope of 2(2x) = 4x
  • The slope of the line 3x is 3

     Another example for you:

    Example: y = x3 − 6x2 + 12x − 5

    The derivative is: y' = 3x2 − 12x + 12

    The second derivative is: y'' = 6x − 12

    And 6x − 12 is negative up to x = 2, positive from there onwards. So:

    f(x) is concave downward up to x = 2

    f(x) is concave upward from x = 2 on

    And the inflection point is at x = 2:

Point of Inflection - Differentiation, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

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FAQs on Point of Inflection - Differentiation, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

1. What is a point of inflection in mathematics?
Ans. A point of inflection is a point on a graph where the curvature changes sign. It is a point where the concavity of a function changes from being concave up to concave down, or vice versa.
2. How can we identify a point of inflection in a graph?
Ans. To identify a point of inflection in a graph, we need to find the second derivative of the function. If the second derivative changes sign at a particular point, then that point is a potential point of inflection. To confirm if it is indeed a point of inflection, we can check the behavior of the function on both sides of that point.
3. What is the significance of points of inflection in business mathematics?
Ans. Points of inflection are significant in business mathematics as they help in analyzing the behavior of functions and making decisions based on their characteristics. For example, in cost analysis, a point of inflection can indicate a change in the rate of cost increase or decrease, which can impact pricing or production decisions.
4. Can a function have multiple points of inflection?
Ans. Yes, a function can have multiple points of inflection. The number of points of inflection depends on the complexity and behavior of the function. Functions with higher degrees or more complex behavior are more likely to have multiple points of inflection.
5. How are points of inflection related to statistical analysis?
Ans. In statistical analysis, points of inflection can be used to identify critical points or transitions in data patterns. They can help in detecting changes in trends, such as shifts in market demand or consumer behavior. By analyzing the points of inflection, statisticians can make predictions and recommendations for business strategies and decision-making.
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