Relative Extrema

The relative extrema of a function indicate the behavior of the function and tell the points where the function has maxima or minima. Points of relative extrema can be obtained using the second derivative test by checking the sign of the second derivative at the critical points or using the first derivative test by checking the change of sign of the first derivative of the function in the nearest neighborhood of the critical point. First Derivative Test and Second Derivative Test are the two most commonly used methods to determine the relative extrema.

In this article, we will learn how to find the relative extrema of a function using the derivative tests and using graphs. We will also solve different examples to understand the steps for finding the relative extrema and for the clarity of the concept.

Relative extrema are the input values of a function f(x) where f(x) has minimum or maximum values. They can be of two types - relative maxima and relative minima. Graphically, relative extrema are the peaks and valleys of the graph of a function, peaks being the points of relative maxima and valleys being the points of relative minima. The combination of relative maxima and minima is called the relative extrema.

The relative extrema of a function are points on the graph of a function at which the minimum or maximum values of the function are obtained in some interval of the domain of the function. Let us go through the definitions of relative maxima and relative individually:

• Relative Maxima - A point x = b is said to be the point of relative maxima for f(x) if in the 𝛿-neighborhood of b i.e in (b−𝛿, b+𝛿) where 𝛿 can be made arbitrarily small, f(x) < f(b) for all x ∈ (b−𝛿, b+𝛿)∖{b}. In other words, if we consider a small region (interval) around x = b, f(b) should be the maximum in that interval.
• Relative Minima - A point x = a is said to be point of relative minima for f(x) if in the neighbourhood of a, i.e. in (a−𝛿,a+𝛿), where 𝛿 can be made arbitrarily small, f(x) > f(a) for all x ∈ (a−𝛿,a+𝛿)∖{a}. In simple words, if we consider a small interval around x = a, f(a) should be the minimum in that interval.

Please note that a function f(x) can have more than one relative extrema. On the other hand, there can only be one absolute extremum (one absolute maximum and one absolute minimum) of the function over the entire domain.

We can evaluate the relative extrema of a function using derivative tests. There are two tests, namely:

• The first derivative test
• The second derivative test

We will study the two tests in detail with the help of examples to understand their applications. In the first derivative test, we check the sign of the first derivative as move via the critical points, and in the second derivative test, we check the sign of the second derivative at the critical points. Let us explore the two tests thoroughly in the following sections.

Now, to find the relative extrema using the first derivative test, we check the change in the sign of the first derivative of the function as we move through the critical points. The slope of the graph of the function is given by the first derivative. Consider a continuous differentiable function f(x) with a critical point at x = c such f'(c) = 0. Then, we have the following conditions for the points of relative extrema:

• Relative Maxima: If the sign of f'(x) changes from positive to negative as we move from left to right through the point x = c, i.e., if f'(x) > 0 for values of x in left 𝛿-neighborhood of c, and f'(x) < 0 for values of x in right 𝛿-neighborhood of c, where 𝛿 can be arbitrarily small, then x = c is a point of relative maxima.
• Relative Minima: If the sign of f'(x) changes from negative to positive as we move from left to right through the point x = c, i.e., if f'(x) < 0 for values of x in left 𝛿-neighborhood of c, and f'(x) > 0 for values of x in right 𝛿-neighborhood of c, where 𝛿 can be arbitrarily small, then x = c is a point of relative minima.
• Test Fails: If the sign of the first derivative of f(x) does not change we move through the point c, then x = c is called the point of inflection.

Let us consider an example to understand how to find the points of relative extrema using the first derivative test step-wise. For this, consider the function f(x) = 2x³ - 3x² + 6. Now, follow the given steps to find its points of relative extrema:

Step 1: Determine the derivative of f(x)

f'(x) = 6x² - 6x

Step 2: Equate the derivative to 0, i.e., f'(x) = 0 to find the critical points.

f'(x) = 0

⇒ 6x² - 6x = 0

⇒ 6x(x - 1) = 0

⇒ x = 0, or x = 1

Therefore, x = 0 and x = 1 are the critical points.

Now, to determine the points of relative extrema, we will consider points on the left and right sides of these critical points.

Step 3: Find a point on the left side and right side of the critical points and check the value of the derivative at these points.

Consider x = -1 on the left side and x = 1/2 on the right side of the critical point x = 0 and check the value of f'(x) at these points.

f'(-1) = 6(-1)² - 6(-1) = 6 + 6 = 12 > 0

f'(1/2) = 6(1/2)² - 6(1/2) = 6/4 - 6/2 = 3/2 - 3 = -3/2 < 0

Since the value of f'(x) changes from positive to negative, therefore x = 0 is a point of relative maxima.

Similarly, consider x = 1/2 on the left side and x = 2 on the right side of the critical point x = 1 and check the value of f'(x) at these points.

f'(1/2) = 6(1/2)² - 6(1/2) = 6/4 - 6/2 = 3/2 - 3 = -3/2 < 0

f'(2) = 6(2)² - 6(2) = 24 - 12 = 12 > 0

Since the value of f'(x) changes from negative to positive, therefore x = 1 is a point of relative minima.

To determine the relative maximum and minimum values, we can find the values of f(0) and f(1), respectively.

Next, to find the points of relative extrema using the second derivative test, we check the sign of the second derivative of the function at the critical points. Generally, if the first derivative test fails, then we use the second derivative test to find the points of relative extrema. Consider a function f(x) that is differentiable twice and a critical point x = c within the domain of f(x) such that f'(c) = 0, then we have the following conditions:

• If f''(c) < 0, then x = c is a point of relative maxima.
• If f''(c) > 0, then x = c is a point of relative minima.
• The test fails if f''(c) = 0. In this case, x = c is called the point of inflection.

Let us consider an example to understand how to find the points of relative extrema using the second derivative test step-wise. For this, consider the function f(x) = 2x³ + 3x² - 12x + 5. Now, follow the given steps to find its points of relative extrema:

Step 1: Determine the derivative of f(x)

f'(x) = 6x² + 6x - 12

Step 2: Equate the derivative to 0, i.e., f'(x) = 0 to find the critical points.

f'(x) = 0

⇒ 6x² + 6x - 12 = 0

⇒ 6(x² + x - 2) = 0

⇒ x² + x - 2 = 0

⇒ x² + 2x - x - 2 = 0

⇒ x(x + 2) - 1(x + 2) = 0

⇒ (x - 1) (x + 2) = 0

⇒ x = 1, or x = -2

Therefore, x = -2 and x = 1 are the critical points.

Step 3: Determine the second derivative of f(x).

f''(x) = 12 x + 6

Step 4: Substitute the critical points into f''(x) and check the sign of the second derivative.

f''(-2) = 12(-2) + 6 = -24 + 6 = -18 < 0 ⇒ x = -2 is a point of relative maxima.

f''(1) = 12(1) + 6 = 12 + 6 = 18 > 0 ⇒ x = 1 is a point of relative minima.

We have learned to determine the points of relative extrema algebraically using the derivative tests. Next, we will learn to identify the relative extrema of a function using a graph. Peaks and valleys in a graph indicate the relative extrema of the function. As we can see in the graph below, there are valleys at x = a and x = c, and the function has minimum values at these points, hence x = a and x = c are the points of relative minima. Similarly, we see peaks at x = b and x = d in the graph. The function has maximum values at these points, and hence x = b and x = d are the points of relative maxima.

Important Notes on Relative Extrema

• A function can have more than one relative extrema but there can only be one absolute maximum and absolute minimum point.
• The value of x within the domain of f(x), which is neither a local maximum nor a local minimum, is called the point of inflection.

Related Topics on Relative Extrema

• Relative Maxima
• First Derivative Test
• Second Derivative Test
• Application of Derivatives

What is Relative Extrema in Calculus?

Relative extrema are the input values of a function f(x) where f(x) has minimum or maximum values. They can be of two types - relative maxima and relative minima.

What are Relative Extrema on a Graph?

Peaks and valleys in a graph indicate the relative extrema of the function, peaks being the points of relative maxima and valleys being the points of relative minima.

When is There No Relative Extrema?

Relative extrema do not occur at the endpoints of the domain of the function.

How to Find Relative Extrema?

We can find relative extrema using a graph. They can also be determined algebraically using the first and second derivative tests.

How to Find Relative Extrema Using the Second Derivative Test?

Consider a function f(x) that is differentiable twice and a critical point x = c within the domain of f(x) such that f'(c) = 0, then we have the following conditions:

• If f''(c) < 0, then x = c is a point of relative maxima.
• If f''(c) > 0, then x = c is a point of relative minima.
• The test fails if f''(c) = 0. In this case, x = c is called the point of inflection.

What is First Derivative Test for Relative Extrema?

For the first derivative test to find relative extrema, consider a continuous differentiable function f(x) with a critical point at x = c such f'(c) = 0. Then, we have the following conditions for the points of relative extrema:

• Relative Maxima: If the sign of f'(x) changes from positive to negative as we move from left to right through the point x = c, i.e., if f'(x) > 0 for values of x in left 𝛿-neighborhood of c, and f'(x) < 0 for values of x in right 𝛿-neighborhood of c, where 𝛿 can be arbitrarily small, then x = c is a point of relative maxima.
• Relative Minima: If the sign of f'(x) changes from negative to positive as we move from left to right through the point x = c, i.e., if f'(x) < 0 for values of x in left 𝛿-neighborhood of c, and f'(x) > 0 for values of x in right 𝛿-neighborhood of c, where 𝛿 can be arbitrarily small, then x = c is a point of relative minima.
• Test Fails: If the sign of the first derivative of f(x) does not change we move through the point c, then x = c is called the point of inflection.

What is the Difference Between Absolute and Relative Extrema?

A function can have one absolute extremum (one absolute maximum and one absolute minimum). On the other hand, a function can have as many relative extrema as possible. Also, absolute extrema are one of the relative extrema only.

Is a Point of Inflection a Relative Extrema?

A point of inflection may or may not be a relative extremum. For example, consider (x) = x² for x ≤ 0. Here, x = 0 is an inflection point using the first derivative test but x = 0 is also a point of relative minima.