Calculating the critical point of a function can be a bit tricky if you are new to it. Several steps need to be followed, and certain considerations observed to arrive at the right conclusions. Unlike other mathematical calculations that are pretty straightforward, performing the mathematical computation of a function’s critical points is only one of the several ways that you could arrive at your answer. However, before we go further, it is important to explain what Critical points are.
A Critical Point is actually one of the different points in the domain of a function for which the function derivative cannot be found or is equal to zero. The derivative being talked about is the first-order derivative of the function.
In essence, any point of a function F(x) that makes its derivative
is a critical point as long as it is included in the function’s domain.
The above means that a number can only be a critical point if it is in the domain of a function and can make the derivative of the same function either impossible to determine or equal to zero.
There are different types of functions for which critical points can be computed, and this includes complex functions (functions whose domains are complex numbers). However, the scope of this article is to provide students with the knowledge of how to calculate the Critical points of Real Functions (functions whose domain are real numbers)
There are several steps to calculating the critical points of a function, and as was stated already, there are several methods to find critical points. However, you should understand that the most comprehensive computation is the mathematical form. The graph method is also reliable. However, there is barely a graph of the original function in most cases, and mathematical computation becomes the preferable option.
Critical Point Formula
Let F(x) be a function with at least one point in its domain. If there is a point or points from its domain where F(x) becomes impossible to differentiate i.e
Before further calculating critical points, we will consider a very important part of the concept, which is critical numbers.
Critical Numbers Vs Critical Points
In many instances, people tend to replace critical points with critical numbers. That is, it is common to see questions where you may be asked to find the critical numbers of a function instead of the critical points of a function. Mathematically, these two terms are very different and connote different meanings in Critical Points computation.
Recall that a function’s critical points are those points in its domain that make it impossible for it to be differentiated or ensure its eventual derivative equals zero. Now you need to understand that these points are numbers. When you calculate a function and find out the numbers that make it derivate zero or impossible to find, these numbers are known as critical Numbers and not critical points. The numbers can only be confirmed to be critical points if they are tested on a real number line and proven to change the function directly in a graph.
In summary, critical numbers are numbers that have the potential to become critical points. When they are fully confirmed to be in the function domain and can change its direction, they become Critical Points. So while all critical points are critical numbers, not all critical numbers will become critical points.
A number of examples will be considered in this article to explain the difference further.
How to Calculate Critical Points Using the Graph Method
The graphical method of calculating the critical point is preferable when the original graph showing the function f(x) behaviour is available.
Below are the steps to compute Critical Points based on a graph
Step 1: Look out for points in the curve where the direction of the graph’s curve changes.
Step 2: Mark out those points and trace them down to the real numbers covered by the entire curve. These numbers are the critical points
Step 3: Any direction changing point that is traced to a number outside the function domain (that is, outside the curve border) is only a critical number and cannot be a critical point
How to Calculate Critical Points by Derivative
Below are the steps to compute the Critical points by derivatives.
Step 1: Differentiate the function
Step 2: equate the function derivate to zero and calculate for x. the value or values computed will be the critical number or numbers.
Step 3: Test whether the critical numbers are critical numbers to know whether they change the direction of the function. To do that, you have to follow the steps below
Step 3a: construct a real number line around the solved Critical Points. These real numbers must be very close to the critical number and assumed to be part of the function domain if there is no graph to tell the function domain.
Step 3b: Try the critical numbers on the derivative and do the same with the surrounding real numbers constructed. If the surrounding real numbers before a critical number are positive values while those after it are negative values, then the critical number is a critical point. This will also be the case if there is a reverse of signs
Note: The negative and positive signs are used to identify the critical points because they represent a change in direction. The moment the critical numbers influence a change in the signs of the number after it compared to those before it, then it means it is the point at which there is a change in the original function graph
If after a critical number, the signs of the number after it are the same as the same signs of the domain numbers before it, then it is not a critical point.
The Critical Points Graph
As already stated in this article, a Critical point is a point in which a function change direction. We will need to use a graph to know if a function changes direction when it gets to a particular point. Now, let the graph below represent a function F(x) where the number line at the bottom is the function’s domain and possible critical numbers.
From the graph above, there are three different points where the direction changed: critical numbers and potential critical points. Before getting to the orange point which is (point 0 from the graph) the direction of the function F(x) was upward. Also, when it got to the black point, the direction changed again. The critical number is 2 for this black point because that is where the direction changes.
There is again another change at the blue pint, and that gives us another critical number which is 6
Therefore, 0, 2 and 6 are critical numbers from the graph. To know if they are possible critical points, they will need to meet the second condition on whether they are in the function’s domain. Obviously, we can see that the curve borders over all the possible critical numbers as it starts from -1 and ends at 8. So obviously, the three critical numbers are critical points
This particular method of finding critical points is the graph method.
In most cases, though, the graph method will not be convenient because most functions are in expressions than graphs. In that case, the best option is to use the method of differentiation.
Find the critical points of the function
Step 1: Differentiate the function
Differentiating the function by the power rule will give
Step 2: equate the function derivate to zero and calculate for x. the value or values computed will be the critical point or points.
This means that we will make
Now, solving for the second x, which is an expression (2)
2 = -x
X = -2
So x = -2 is another critical number
In summary, the critical numbers are x = -2, and x = 0
Step 3: Test whether the critical numbers are critical numbers to know whether they change the direction of the function.
To test the critical numbers x = -2, and X = 0
We will construct a surrounding number around them. To do this, we will simply add two real numbers lesser than both on the left side and two real numbers higher than both on the right side. These numbers will be assumed to be part of the function domain.
So since the critical numbers are
Adding those surrounding numbers
-4, -3, -2, -1, 0, 1, 2
-4, -3, are the two real numbers lesser than -2 and 0 added to the left side of the real number system.
1, 2, are the two real numbers greater than -2 and 0, added to the right side of the real number system
The -1 between the critical numbers -2, and 0 is a natural addition to make the real number system logical.
Step 3b: Try all the numbers on the derivative.
Since the derivative is
Since the positive signs after -2, a critical number changed to negative, then -2 influences a change in trend (direction). As such, it is a critical point.
Since the negative sign after 0 a critical number changes to positive signs, then -1 influences a change in trend (direction). As such, it is a critical point.
Computing critical points require more logic than mathematical complexity. You have to understand the steps properly to fully identify the points, and that is what this article has tried to achieve. The steps for computing are quite easy, and you only need to follow them one at a time.
We considered two examples, with the first outlining how to use the graphical method to find critical points and the second how to use mathematical computations. Both methods have Instances where they are relevant, and you should employ them likewise.
Also, as stated already, the critical point is a vast subject. However, knowing how to solve these examples will go a long way to help you understand them better.