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Maxima and Minima

Maxima and minima are known as the extrema of a function. Maxima and minima are the maximum or the minimum value of a function within the given set of ranges. For the function, under the entire range, the maximum value of the function is known as the absolute maxima and the minimum value is known as the absolute minima.

There are other maxima and minima of a function, which are not the absolute maxima and minima of the function and are known as local maxima and local minima. Let us learn more about local maxima and minima, absolute maxima and minima, and how to find the maxima and minima of the function.

1.	Maxima and Minima of a Function
2.	Local Maxima and Minima
3.	Absolute Maxima and Minima
4.	How to Find Maxima and Minima of a Function?
5.	FAQs on Maxima and Minima

What are Maxima and Minima of a Function?

Maxima and minima are the peaks and valleys in the curve of a function. There can be any number of maxima and minima for a function. In calculus, we can find the maximum and minimum value of any function without even looking at the graph of the function. Maxima will be the highest point on the curve within the given range and minima would be the lowest point on the curve.

The combination of maxima and minima is extrema. In the image given below, we can see various peaks and valleys in the graph. At x = a and at x = 0, we get maximum values of the function, and at x = b and x = c, we get minimum values of the function. All the peaks are the maxima and the valleys are the minima.

maxima and minima

There are two types of maxima and minima that exist in a function, which are:

Local Maxima and Minima
Absolute or Global Maxima and Minima

Let us learn about them in detail.

Local Maxima and Minima

Local maxima and minima are the maxima and minima of the function which arise in a particular interval. Local maxima would be the value of a function at a point in a particular interval for which the values of the function near that point are always less than the value of the function at that point. Whereas local minima would be the value of the function at a point where the values of the function near that point are greater than the value of the function at that point.

Local Maxima: A point x = b is a point of local maximum for f(x) if in the 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}. This simply means that if we consider a small region (interval) around x = b, f(b) should be the maximum in that interval.
Local Minima: A point x = a is a point of local minimum for f(x) if in the neighbourhood of a, i.e. in (a−𝛿,a+𝛿), (where 𝛿 can have arbitrarily small values), f(x) > f(a) for all x ∈ (a−𝛿,a+𝛿)∖{a}. This means that if we consider a small interval around x = a, f(a) should be the minimum in that interval.

In the image given below, we can see that x = b and x = d, are the local maxima, and x = a and x = c, are the local minima.

local maxima and minima

Absolute Maxima and Minima

The highest point of a function within the entire domain is known as the absolute maxima of the function whereas the lowest point of the function within the entire domain of the function, is known as the absolute minima of the function. There can only be one absolute maximum of a function and one absolute minimum of the function over the entire domain. The absolute maxima and minima of the function can also be called the global maxima and global minima of the function.

Absolute maxima: A point x = a is a point of global maximum for f(x) if f(x) ≤ f(a) for all x∈D (the domain of f(x)).
Absolute minima: A point x = a is a point of global minimum for f(x) if f(x) ≥ f(a) for all x∈D (the domain of f(x)).

In the image given below, point x = a is the absolute maxima of the function and at x = b is the absolute minima of the function.

absolute maxima and minima

How to Find Maxima and Minima of a Function?

Maxima and minima of a function can be calculated by using the first-order derivative test and second-order derivative test. Derivative tests are the quickest ways to find the maxima and minima of a function. Let us discuss them one by one.

First Order Derivative Test for Maxima and Minima

The first derivative of a function gives the slope of the function. Near a maximum point, the slope of the curve increases as we go towards the maximum point then becomes 0 at the maximum point and then decreases as we move away from the maximum point. Similarly, near the minimum point, the slope of the function decreases as we move towards the minimum point then becomes 0 at the minimum point, and then increases as we move away from the minimum point. We use this information to know whether the point is maxima or minima.

Let say we have a function f which is continuous at the critical point, defined in an open interval I and f’(c) = 0 (slope is 0 at c). Then we check the value of f'(x) at the point left to the curve and right to the curve and check the nature of f'(x), then we can say, that the given point will be:

Local maxima: If f’(x) changes sign from positive to negative as x increases via point c, then f(c) gives the maximum value of the function in that range.
Local minima: If f’(x) changes sign from negative to positive as x increases via point c, then f(c) gives the minimum value of the function in that range.
Point of inflection: If the sign of f’(x) doesn’t change as x increases via c, and the point c is neither the maxima nor minima of the function, then the point c is called the point of inflection.

Second-Order Derivative Test for Maxima and Minima

In the second-order derivative test for maxima and minima, we find the first derivative of the function and if it gives the value of the slope equal to 0 at the critical point x = c (f’(c) = 0), then we find the second derivative of the function. If the second derivative of the function exists within the given range, then the given point will be:

Local maxima: If f''(c) < 0
Local minima: If f''(c) > 0
Test fails: If f''(c) = 0

Important Notes on Maxima and Minima:

Maxima and minima are the peaks and valleys in the curve of a function.
There can only be one absolute maxima of a function and one absolute minimum of the function over the entire domain.
A function f is called a monotonous function in the interval I, if f is either increasing in I or decreasing in I.

Related Topics on Maxima and Minima:

Solved Examples on Maxima and Minima

Example 1: Find the points of maxima and minima of a function:y = 2x³ - 3x² + 6

Solution

Given function: y = 2x³ - 3x² + 6

Using second order derivative test for the maxima and minima of a function:

Taking first order derivative of:

y = 2x³ - 3x² + 6 ----- (eq 1)

Differentiate both of sides (eq 1), w.r.t x.

⇒ dy/dx = d(2x³)/dx - d(3x²)/dx + d(6)/dx

⇒ dy/dx = 6x² - 6x + 0

⇒ dy/dx = 6x² - 6x ------ (eq 2)

Putting dy/dx = 0 to find critical points,

⇒ 6x² - 6x = 0

⇒ 6x (x - 1) = 0

⇒ x = 0,1

The critical points are 0 & 1.

Differentiate both of sides of (eq 2), w.r.t x.

⇒ d²y/dx² = d(6x²)/dx - d(6x)/dx

⇒ d²y/dx² = 12x - 6

Now, put the values of x and find the max or min value.

At x = 0, d²y/dx² = 12(0) - 6 = -6 < 0, hence x = 0 is a point of maxima

At x = 1, d²y/dx² = 12(1) - 6 = 6 > 0, hence x = 1 is a point of minima

Answer: The maxima of the function is at x = 0 and minima of the function is at x = 1.
Example 2: Find the extrema of the given function: f(x) = -3x² + 4x + 7 and the extremum value using maxima and minima formulas.

Solution:

Using second order derivative test for the maxima and minima of a function:

Given function: f(x) = -3x² + 4x + 7 ----------------(eq 1)

Differentiate on both sides of (eq 1), w.r.t x.

⇒ dy/dx = d(-3x²)/dx + d(4x)/dx + d(7)/dx

⇒ dy/dx = - 6x + 4

Putting dy/dx = 0 to find critical points.

⇒ -6x + 4 = 0 ----------------(eq 2)

⇒x = 2/3

The critical point is 2/3.

Differentiate both sides of (eq 2), w.r.t x.

⇒ d²y/dx² = d(-6x)/dx + d(4)/dx

⇒ d²y/dx² = -6

Since d²y/dx²< 0, the given curve will have maxima at x = 2/3.

The maxima value of f(x) at x = 2/3 is,

f(2/3) = -3(2/3)² + 4(2/3) + 7 = -4/3 + 8/3 + 7 = 25/3

Answer: The maxima of the function is at x = 2/3 and maximum value is 25/3.

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Maxima and Minima Practice Problems

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FAQs on Maxima and Minima

What are Maxima and Minima of a Function?

Maxima and minima of a function are the points where the value of the function is maximum or minima within the delta neighborhood of the point.

What is Local Maxima and Minima?

Local maxima and minima are the maxima and minima of the function that arises in a particular interval. Local maxima would be the point in the particular interval for which the values of the function near that point are always less than the value of the function at that point. Whereas local minima would be the point where the values of the function near that point are greater than the value of the function at that point.

What is Absolute Maxima and Minima?

The highest point of a function within the entire domain is known as the absolute maxima of the function whereas the lowest point of the function within the entire domain of the function, is known as the absolute minima of the function. The can only be one absolute maximum of a function and one absolute minimum of the function over the entire domain.

How to Find Local Maxima and Minima on the Graph?

Maxima and minima are the peaks and valleys in the curve of a function. There can be any number of maxima and minima for a function. In the graph of a function, if we want to find the local maxima and minima, we just look for the peaks and valleys in the graph. All the peaks will be our local maxima and valleys will be the local minima.

How to Find Maxima and Minima Algebraically?

What is the First Derivative Test for Maxima and Minima of a Function?

The first derivative of a function gives the slope of the function. According to the first derivative test, near a maximum point, the slope of the curve increases as we go towards the maximum point then becomes 0 at the maximum point and then decreases as we move away from the maximum point. Similarly, near the minimum point, the slope of the function decreases as we move towards the minimum point then becomes 0 at the minimum point, and then increases as we move away from the minimum point. We use this information to know whether the point is maxima or minima.

What is the Second Order Derivative Test for the Maxima and Minima of a Function?

In the second-order derivative test, we first find the first derivative of the function and if it gives the value of the slope equal to 0 (f’(c) = 0), then we find the second derivative of the function. If the second derivative of the function exists within the given range, then the given point will be:

Local maxima: If f”(c) < 0
Local minima: If f”(c) > 0
Test fails: If f”(c) = 0

Can There be 2 Absolute Maximas in Maxima and Minima?

The highest point of a function within the entire domain is known as the absolute maximum of the function whereas the lowest point of the function within the entire domain of the function, is known as the absolute minimum of the function. There can only be one absolute maximum of a function and one absolute minimum of the function over the entire domain.

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