Meaning of Minimum
The minimum is the smallest or least quantity that is possible or necessary. You will not be fired if you do the bare minimum of work at your job, but you will not be promoted.
Since minimum is Latin for smallest, English speakers have obviously done the least amount of interfering with the sense of this term. Obviously, the opposite is Maximum. At the minimum, you should understand that the term refers to something's smallest limit.
Meaning of Min (Minimum)
In a given set of data, the smallest or the least number is said to be the minimum number. It can be easily detected by arranging the given data in descending or ascending order. For example, the given series of numbers is 167, 897, 69, 301, 999, 294. If we arrange this series in descending order we will get 999, 897, 301, 294, 167, 69. Thus, this clearly shows 69 is the minimum number which got its place at the end.
Minimum in Mathematics
The maxima & minima (the plurals of maximum and minimum) of the function, known collectively as extrema (the plural of extremum), are the function's largest and smallest values, either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to introduce a general technique for determining the maxima and minima of functions, known as adequality.
The maximum and minimum of a set are the greatest and least elements in the set, as defined by set theory. There is no minimum or limit for unbounded infinite sets, such as the set of real numbers.
Maxima or Minima of a Function
When the domain of a function in which an extremum is to be found is made up entirely of functions (i.e. if the extremum of a function is to be found), the extremum is found using the calculus of variations.
A maximum is a term used to describe a high point (plural maxima). A minimum is a term used to describe a low point (plural minima). Extremum is a general term for maximum or minimum (plural extrema). When there are higher (or lower) points elsewhere but not nearby, we say local maximum (or minimum).
In Relation to Sets
Sets may also have maxima and minima specified. If an ordered set S has the greatest element m, then m is the set's maximum element, also known as max (S). In addition, if S is a subset of an ordered set T and m is the greatest element of S with (respect to the order induced by T), then m is the least upper bound of S in T. The least element, minimal element, and greatest lower bound all yield similar results. Since the maximum (or minimum) of a set can be computed from the maxima of a partition, the maximum (or minimum) of a set can be computed quickly in databases; formally, they are self-decomposable aggregation functions.
Absolute Minimum and Maximum
An absolute maximum point is a point at which the function achieves its maximum value. An absolute minimum point, on the other hand, is the point at which the function takes on its smallest possible value.
If you already know how to locate relative minima and maxima, you'll need to consider the ends in both directions to find absolute extrema points.
Conclusion
At the minimum, you should be able to understand that the smallest number is said to be the minimum number. A maximum is a term used to describe a high point (plural maxima). A minimum is the smallest or least quantity that is possible or necessary. In mathematics, if an ordered set S has the greatest element m, then m is the set's maximum element, also known as max (S) absolute maximum points are self-decomposable aggregation functions. An absolute minimum point is a point at which the function achieves its maximum value. If you already know how to locate relative minima and maxima, you'll need to consider the ends in both.
FAQs on Minimum
1. What is the minimum of a set of numbers?
In mathematics, the minimum of a set of numbers is simply the smallest number in that set. To find it, you can arrange the numbers in ascending order; the first number in the sequence will be the minimum. For example, in the set {15, 8, 29, 4}, the minimum is 4.
2. What is the minimum value of a function?
The minimum value of a function is the lowest y-value that the function reaches within a given interval or across its entire domain. This point on the graph is often referred to as a minimum point or an extremum. It represents the smallest possible output of the function.
3. What is the difference between a local minimum and an absolute minimum?
The distinction between a local and absolute minimum is crucial in calculus.
- A local minimum is a point on a function's graph that is lower than all other points in its immediate vicinity, but there might be a lower point elsewhere on the graph. A function can have multiple local minima.
- An absolute minimum is the single lowest point across the function's entire domain. It is the smallest value the function ever attains. A function can only have one absolute minimum value.
4. How do you find the minimum of a function using derivatives?
As per the CBSE syllabus for Class 12, the first and second derivative tests are used to find the minimum of a function. The typical steps are:
- Step 1: Find the first derivative of the function, f'(x).
- Step 2: Set the first derivative to zero (f'(x) = 0) and solve for x to find the critical points.
- Step 3: Find the second derivative, f''(x).
- Step 4: Substitute each critical point 'c' into the second derivative. If f''(c) > 0, the function has a local minimum at x = c.
5. Why is the second derivative test used to confirm a minimum?
The second derivative test works because it describes the concavity of the function's graph. A positive second derivative (f''(x) > 0) at a critical point indicates that the graph is concave up (shaped like a 'U') at that point. At the bottom of a 'U' shape, there is always a minimum, which is why this test reliably confirms a minimum point.
6. What are some real-world examples of using the minimum concept?
The concept of finding a minimum is widely applied to optimise real-world problems. For instance:
- Business and Economics: Companies use it to find the production level that results in the minimum cost per unit.
- Engineering: It helps in designing structures, like bridges or containers, using the minimum amount of material while maintaining strength.
- Logistics: It is used to calculate the shortest path between two points to minimise travel time or fuel consumption.
7. Can a function have a minimum value but no maximum value?
Yes, a function can have a minimum value without having a maximum value. A common example is the parabola f(x) = x². This function has an absolute minimum value of 0 at the point x=0. However, the function's arms extend upwards to infinity, so there is no single highest point, and therefore, it has no maximum value.
