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Page 1
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 ?nd the maxima
and minima of the function.
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 ?nd 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 extreme. 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.
Page 2
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 ?nd the maxima
and minima of the function.
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 ?nd 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 extreme. 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 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.
Page 3
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 ?nd the maxima
and minima of the function.
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 ?nd 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 extreme. 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 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- ?? , b+ ?? ) where ?? can be made arbitrarily small, f(x) < f(b) for all x ? (b- ?? , b+ ?? ) \{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- ?? ,a+ ?? ), (where ?? can have arbitrarily small values), f(x) > f(a) for all x ?
(a- ?? ,a+ ?? ) \{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
Page 4
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 ?nd the maxima
and minima of the function.
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 ?nd 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 extreme. 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 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- ?? , b+ ?? ) where ?? can be made arbitrarily small, f(x) < f(b) for all x ? (b- ?? , b+ ?? ) \{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- ?? ,a+ ?? ), (where ?? can have arbitrarily small values), f(x) > f(a) for all x ?
(a- ?? ,a+ ?? ) \{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 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.
Page 5
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 ?nd the maxima
and minima of the function.
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 ?nd 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 extreme. 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 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- ?? , b+ ?? ) where ?? can be made arbitrarily small, f(x) < f(b) for all x ? (b- ?? , b+ ?? ) \{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- ?? ,a+ ?? ), (where ?? can have arbitrarily small values), f(x) > f(a) for all x ?
(a- ?? ,a+ ?? ) \{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 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.
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
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FAQs on Maxima and Minima - Business Mathematics and Statistics - B Com
| 1. What are the basic concepts of maxima and minima in calculus? |  |
Ans. Maxima and minima are concepts in calculus that refer to the highest and lowest points of a function, respectively. A maximum point is where the function reaches its highest value in a given interval, while a minimum point is where it reaches its lowest value. These points are typically found using the first derivative test and the second derivative test, which help identify critical points and determine their nature.
| 2. How do you find the maxima and minima of a function using derivatives? | |
Ans. To find the maxima and minima of a function, you first take the derivative of the function and set it equal to zero to find critical points. Then, you can use the second derivative test: if the second derivative at a critical point is positive, the point is a local minimum; if it is negative, the point is a local maximum. If the second derivative is zero, further analysis is needed to determine the nature of the critical point.
| 3. What is the significance of the first and second derivative tests in finding extrema? | |
Ans. The first derivative test is significant because it helps identify critical points where the function changes from increasing to decreasing or vice versa. This allows us to classify those points as maxima or minima. The second derivative test provides further confirmation by indicating the concavity of the function at those critical points, thus helping to ascertain whether they are indeed maxima or minima.
| 4. Can maxima and minima occur at endpoints of an interval? | |
Ans. Yes, maxima and minima can occur at the endpoints of a closed interval. When determining the absolute maximum or minimum of a function over a closed interval, it is essential to evaluate the function at the critical points found within the interval as well as at the endpoints. The largest value among these evaluations will be the absolute maximum, and the smallest will be the absolute minimum.
| 5. What are some practical applications of maxima and minima in real life? | |
Ans. Maxima and minima have various practical applications in real life, such as in economics for maximizing profit or minimizing cost, in engineering for optimizing design and material usage, and in physics for finding optimal conditions for motion and forces. These concepts are also crucial in fields like statistics, operations research, and environmental science, where optimization problems are common.
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