Maxima and Minima - Using First Derivative Test

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 the 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 a function.

In set theory, the maximum and minimum of a given set are considered as the greatest and least elements of the set respectively whereas the set of real numbers has no maximum and minimum value. We can find maxima and minima using the first derivative test, and second derivative test. In the article, we will discuss how to find maxima and minima using the first derivative test.

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.

Method to Find Whether a Critical point is Maximum, Minimum, Or Neither Using First Derivative Test

The first derivative test is a method to determine whether a critical point is maximum, minimum or neither. 

First Derivative Test: for a Given Critical Point

• 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 a 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 nor minimum.

Example

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.

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The diagram above shows x = 3 is the maximum.

How to Find Maxima and Minima Using First Derivative Test

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. 

Then Observe the Following Points

• 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.

Maxima and Minima Using First Derivative Test Example

Find the critical points and any local maxima or minima of a given function f(x)=1/4x⁴ -8x

Here are the steps:

1.  The first step is to differentiate f(x)= \[ \frac{1}{4x^{4} - 8x} \]

f' (x)= \[ \left [ \frac{1}{4x^{4} - 8x} \right ]' \]= 1/4 ( 4x³ - 8) = x³ - 8

2. 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

3. 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 negative to positive, 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.

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Points to Remember

• Maxima and minima in calculus are calculated by using the concept of derivatives.  

• The concept of the derivatives gives out the information regarding the gradient or slope of the function as the points get located where the gradient is zero.

• These points are known as turning points or stationary points. 

• These points correspond to the largest or smallest values of the function.

• Maxima and Minima are the most common concepts in differential calculus. 

• A branch of Mathematics known as Calculus of Variations deals with the maxima and the minima of the functional. 

• The calculus of variations is concerned with the variations in the functional during which a  small change in the function leads to the change in the functional value.

• The first variation is defined as the linear part of the change in the functional. 

• The second part of the variation is known in the quadratic part. Functional is expressed as the definite integrals that involve the functions and their derivatives.

• The functions that maximize or minimize the functional are to be found using the Euler i.e, Lagrange of the calculus of variations. 

• The two Latin words, ‘maxima’ and ‘minima’ mean the maximum and minimum value of a function respectively. 

• The maxima and minima are collectively known as the “Extrema”. 

• The Critical point of a differential function of a complex or real variable is any value in its domain where its derivative is 0. 

• It can be inferred that every local extremum is a critical point, however, every critical point does not have to be a local extremum. 

• If there is a function that is continuous, it must have maxima and minima or local extrema. 

• It implies that all such functions will have critical points. 

• If the given function is monotonic, the maximum and minimum values lie at the endpoints of the domain of the definition of that function.

• Maxima and minima are, therefore, very important concepts in the calculus of variations, which helps to find the extreme values of a function. 

• One can use the two values and where they occur for a function using the first derivative method or the second derivative method.

• If f(x) is a continuous function in its domain, then one maximum or one minimum should lie between equal values of f(x).

• Maxima and minima occur alternately which is, between two minima then there is one maximum and vice versa. 

• If f(x) tends to be infinity as x tends to a or b and f(x) = 0 only for one value x, that is, c between a and b , then f(c) is the minimum and of the least value.  If f(x) tends to minus infinity as x tends to a or b , then f(c) is of the maximum and of the highest value.