In Mathematics, the maxima and minima (the plural of maximum and minimum respectively) of a given function are collectively known as the extrema ( the plural term of extremum). The two terms maxima and minima are the smallest and largest value of function, either within a given range, or the entire domain. Pierre de Fermat was one of the renowned Mathematicians to introduce a general technique, adequality, for determining the maxima and minima of function.
In set theory, the maximum and minimum of a given set are considered as greatest and least elements of the set respectively whereas the set of real numbers has no maximum and minimum value.
Finding the maxima and minima value is the goal of mathematicians. We can find maxima and minima using the first derivative test, and second derivative text. In the article we will discuss how to find maxima and minima using the first derivative test.
Let us consider f real -valued function, and [ a,b] is an interval on which function f is defined and differentiable. Further, if c is considered as the critical point of f in [a,b], then
If f’(x) > 0 ( greater than 0) for all x < c and f’(x) < 0 ( lesser than 0) for all x > C, then f (c) will be considered as the maximum value of function f in the interval [a,b].
If f’(x) < 0 ( lesser than 0) for all x > c and f’(x) > 0 ( greater than 0) for all x < C, then f (c) will be considered as the minimum value of function f in the interval [a,b].
In simple words, we can say that a point is determined as the maximum of a function if the function increases before and decreases after it whereas a point is considered as the minimum if the function decreases before and increases after it.
The first derivative test is a method to determine whether a critical point is maximum, minimum or neither.
If the derivative is negative on the left side of the critical point and positive on the right side of the critical point, then the critical point is considered as minimum.
If the derivative is positive on the left side of the critical point and negative on the right side of the critical point, then the critical point is considered as maximum.
In any other situation, the critical point will neither be maximum or minimum.
Let us consider f(x)= 6x - x²
The derivative of f(x)= 6x - x² is f'(x)= 6 - 2x
The function f has a critical point at x = 3, as 3 is the solution of 6 - 2x = 0
To determine whether the critical point i.e x = 3 is maximum, minimum , or neither, observe where f is increasing or decreasing.
(Image to be added soon)
The diagram above shows x = 3 is maximum.
The first derivative test is used to determine whether a function is increasing or decreasing on its domain, and to identify its local maxima or minima.
The first derivative test is considered as the slope of the line tangent to the graph at a given point. When the slope is positive, the graph is increasing whereas when the slope is negative, the graph is decreasing. When the slope is 0, the point is considered as a critical point and it can be a local maximum or minimum.
Given a differentiable function, the first derivative test can be applied to determine any local maxima or minima of the given function through the steps given below.
Step 1: Differentiate the given function.
Step 2: Set the derivative equivalent to 0 and solve the equation to determine any critical points.
Step 3: Test the values before and after the critical points to find whether the function that is given is increasing (positive derivative) or decreasing (negative derivative) around the point.
If the first derivative changes from positive to negative at the given point, then the point is determined as a local maximum.
If the first derivative changes from negative to positive at the given point, then the point is determined as a local minimum.
If the first derivative does not change at the given point, then the given point will neither be considered as a local maximum or minimum.
Find the critical points and any local maxima or minima of a given function f(x)= 1/4x² - 8x.
Here are the steps
The first step is to differentiate f(x)= 1/4x⁴ - 8x
f' (x)= [1/4x⁴ - 8x]’ = 1/4. 4x³ - 8 = x³ - 8
The second step is to find the value of x
Let us equate, x³ - 8 = 0
x³ = 8
Hence, the value of x = 2
This implies that, x³ - 8 , has a critical point at x = 2
The third step is to test the points around critical points at x = 1 and x = 3.
For x = 1, f' (x)= 1³ - 8 = 1 - 8 = -7
For x = 3, f' (x)= 3³ - 8 = 27 - 8 = 19
At f' (1)and f' (3), the text point around the critical points changes from positive to negative, this implies a negative slope on the graph of f (x)before the critical point and positive slope on the graph of f'(x)after the critical point (i.e. from left to right). Hence, the critical point x = 2 is a local minimum and can be seen in the graph of f (x)as shown below.
(Image to be added soon)
1. What are the applications of the first derivative test?
First derivative test helps to solve the optimization problem of Economics, Engineering, and Physics. Along with the extreme value theorem, it can be used to determine the absolute maximum and minimum real valued function on a bound and closed interval. Along with the other information such as concavity, asymptotes, and inflection point, it can be used to draw the graph of a function.
2. What is a critical point in maxima and minima?
A critical point of a continuous function (f) with x in its domain has a critical point at that point x if it satisfies the below conditions.
f'(x) = 0
f' (x) = undefined
A critical point of a differentiable function (f) is defined as a point at which derivative is equal to zero.
3. What does the first derivative test tell us about?
The first derivative test tells us about the direction of a function. In other words, it tells us if the function is increasing or decreasing. If the function changes from increasing to decreasing at the point, the function will attain a highest value at that point. Similarly, If the function changes from decreasing to increasing at the point, the function will attain a lowest value at that point. If the function fails to change and remains increasing or remains decreasing, then no highest or lowest point will be attained.