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 of data, 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 or minima of a function, which are not the absolute maxima or 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.
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.
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 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.
Local Maxima: A point x = b is a local maximum for f(x) if in the neighbourhood 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 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.
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 maxima of a function and one absolute minima of the function over the entire domain. The absolute maxima and absolute minima of the function can also be called as the global maxima and global minima of the function.
 Absolute maxima: A point x = a is a 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 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.
How to Find Maxima and Minima of a Function?
Maxima and minima of a function can be calculated by using the firstorder derivative test and secondorder 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
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 the f'(x), then we can say, that the given point will be:
 Local maxima: If f’(x) changes sign from +ve to ve as x increases via point c. And f(c) gives the maximum value of the function in that range.
 Local minima: If f’(x) changes sign from ve to +ve as x increases via point c. And 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 the minima of the function, then the point c is called the point of inflection.
SecondOrder Derivative Test
In the secondorder derivative test, we 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
Important Notes:
 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:
Solved Examples on Maxima and Minima

Example 1: Find the maxima and minima of a function:y = 2x^{3}  3x^{2} + 6
Solution
Given function: y = 2x^{3}  3x^{2} + 6
Using second order derivative test for the maxima and minima of a function:
Taking first order derivative:
y = 2x^{3}  3x^{2} + 6 (eq 1)
Differentiate both of side (eq 1), w.r.t  x.
⇒ dy/dx = d/dx (2x^{3})  d/dx (3x^{2}) + d/dx (6)
⇒ dy/dx = 6x^{2}  6x + 0
⇒ dy/dx = 6x^{2}  6x (eq 2)
Putting dy/dx = 0 to find critical points.
⇒ 6x^{2}  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.
⇒ d^{2}y/dx^{2} = d/dx (6x^{2})  d/dx (6x)
⇒ d^{2}y/dx^{2} = 12x  6
Now, put the values of x and find the max or min value.
At x = 0, d^{2}y/dx^{2} = 12(0)  6 = 6 < 0, hence x = 0 is a maxima
At x = 1, d^{2}y/dx^{2} = 12(1)  6 = 6 > 0, hence x = 1 is a minima
Answer: The maxima of the function is at x = 0 and minima of the curve is at x = 1.

Example 2: Find the extrema of the given function: f(x) = 3x^{2} + 4x + 7
Solution:
Using second order derivative test for the maxima and minima of a function:
Given function: f(x) = 3x^{2} + 4x + 7 (eq 1)
Differentiate on both sides of (eq 1), w.r.t  x.
⇒ dy/dx = d/dx (3x^{2}) + d/dx (4x) + d/dx (7)
⇒ dy/dx =  6x + 4
Putting dy/dx = 0 to find critical points.
⇒ 6x + 4 = 0 (eq 2)
⇒x = 2/3
The critical points is 2/3.
Differentiate both sides of (eq 2), w.r.t  x.
⇒ d^{2}y/dx^{2} = d/dx (6x) + d/dx (4)
⇒ d^{2}y/dx^{2} = 6
Since d^{2}y/dx^{2}< 0, the given curve will have maxima at 2/3
Answer: The maxima of the function is at x = 2/3.
Frequently Asked Questions(FAQs)
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 maxima of a function and one absolute minima 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.
What Does First Derivative Tell you about the Maxima or Minima of a Function?
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.
What Does the Second Order Derivative Tell you About the Maxima or Minima of a Function?
In the secondorder 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?
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 maxima of a function and one absolute minima of the function over the entire domain.