What are critical points in calculus?

The critical points play an important role in solving several equations. Before knowing about critical points in detail, we will see the definition of critical point and then we will see some example to know how to find critical points.

Let’s have critical point definition

For any f(x) (i.e. function of x), if a point x0 exist for which f(x0) exist and either f’(x0) = 0 or f’(x0) does not exist, then x = x0 is called critical point of function f(x). This is how we can define critical point in math.

Learn the way of finding critical points for a function

In last topic, we have learned the critical point definition. Now we will learn about procedure of finding critical points through critical points calculus examples. Below are some examples for critical point calculus which will enhance your idea on critical points of a function. It’s advised that students understand this examples thoroughly to have a sound knowledge on it.

Let’s start with Polynomials function for which you can find critical points easily.
finding critical points

Step 1 : find the derivative of the f(x).

Step 2 : After finding derivative, make solution in simplest form by factorize it. It helps us in finding critical points.
finding critical points

Here f’(x) is polynomial and it exist for all value of x. so for finding critical points we have to find point at which f’(x) = 0.
find point at which f’(x) = 0

So, here we can find three critical points, and they are,
find three critical points

Now we will have another function for finding critical points in the calculus critical points.
critical points example

Step 1 : find the derivative of y.
find the derivative of y

Here y’ exist for all x, so we have to find the point where derivative of y become zero.
 find the point where derivative of y become zero

After solving above equation,
critical points

Note that 2n in the above equation is important. Thus critical points are obtained as given.
critical points are obtained

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