What is Maximum?
In mathematical analysis, the maxima as well as minima of a function, known collectively as extrema (the plural of extremum), are the largest as well as the smallest value of the function, either within a given range (the relative also known as local extrema) or on the entire domain (that is the global or absolute extrema). One of the first mathematicians namely Pierre de Fermat was the first one to propose a general technique, adequality, for finding the maxima of functions as well as minima of functions.
What is Maximum Value?
The maximum value of a function is known to be the place where a function reaches its highest point or reaches the vertex, on a graph. For instance, in this image given below, the maximum value of the function is y equals to the number 5.
Practically, finding the maximum value of any given function can be used to determine maximum profit or can be used to determine the maximum area. This can be very helpful in running a profitable business or any architecture or building project.
How to Determine Maximum Value?
There are three methods for determining the maximum value of a quadratic equation, let’s discuss the three methods. Each of these methods can be used in its unique setting to determine the maximum.
The first way to determine the maximum value is graphing. You will be able to find the maximum value visually by graphing the equation as well as finding the maximum point on the graph. This is quite easy when you have a graphing calculator that can do the work for you. Again, using this graph, you can see that the maximum point of the graph is at y equals 5.
The second way to determine the maximum value is using the equation y equals ax2 + bx + c.
If your equation is in the form ax2 + bx + c, you can easily find the maximum by using the equation:
max = c - (b2 / 4a).
Let’s know the first step, the first step is to determine whether your equation gives a maximum or minimum. This can be easily done by looking at the x2 term. If this term is positive (>0), the vertex point will be a minimum. If it is negative(<0), the vertex will be a maximum.
After determining that you actually will have a maximum point, use the equation to find it. For example, let's find the maximum point of:
-x2 + 4x - 2.
Since the term with the x2 is negative (<0), you know there will be a maximum point. To find it, plug the values into the equation:
max = c - (b2 / 4a)
This will give us:
max = -2 - (42 / 4 * (-1))
4 squared equals 16, and 4 times -1 equals -4. 16 divided by -4 is -4. And, -2 minus -4 becomes -2 plus 4 because two negatives become a positive, so we end up with a maximum value of 2.
There's one more way by which we can determine the maximum value of a function, and that is from the equation:
y = a(x - h)2 + k
As with the last equation, the a term in this equation must be negative (<0) for there to be a maximum. If the a term is negative, the maximum can be found at k. No equation or calculation is necessary - the answer is just k.
Applications of Maxima And Minima in Daily Life-
There are numerous practical applications in which it is desired to find the maximum or minimum value of a particular quantity. Such applications exist in economics, business, and engineering. Many can be solved using the methods of differential calculus described above. For example, in any manufacturing business, it is usually possible to express profit as a function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. In other cases, the shape of a container may be determined by minimizing the amount of material required to manufacture it. The design of piping systems is often based on minimizing pressure drop which in turn minimizes required pump sizes and reduces cost. The shapes of steel beams are based on maximizing strength.
For example, linear programming consists of maximizing (or minimizing) a particular quantity while requiring that certain constraints be imposed on other quantities. The quantity to be maximized (or minimized), as well as each of the constraints, is represented by an equation or inequality. The resulting system of equations or inequalities, usually linear, often contains hundreds or thousands of variables. The idea is to find the maximum value of a particular variable that represents a solution to the whole system.
A practical example might be minimizing the cost of producing an automobile given certain known constraints on the cost of each part, and the time spent by each labourer, all of which may be interdependent. Regardless of the application, though, the key step in any maxima or minima problem is expressing the problem in mathematical terms.
Problems to be Solved
Question 1: Let’s find the maximum of the equation: -3(x - 5)\[^{2}\] - 7
Solution: In the given equation, a = -3, there will be the maximum at -7.
Question 2: Let’s find the maximum of the equation: -5(x - 5)\[^{2}\] + 8
Solution: In the given equation, a = -5, there will be the maximum at 8.
FAQs on Maximum
1. What does the maximum of a function represent in mathematics?
In mathematics, the maximum of a function refers to the point where the function reaches its highest value. This is also known as a 'peak' or 'summit' on its graph. Depending on the scope, it can be a local maximum (the highest point in a specific neighbourhood) or an absolute maximum (the highest point over the function's entire domain). The general term for a maximum or minimum is an extremum.
2. What is the difference between a local maximum and an absolute maximum?
The key difference lies in the range over which the value is considered the highest.
- A local maximum is a point on the graph that is higher than all the other points in its immediate vicinity. A function can have several local maxima.
- An absolute maximum is the single highest point across the entire domain of the function. It is the 'highest of all highs'. While a function can have multiple local maxima, it can only have one absolute maximum value.
3. How can derivatives be used to find the maximum value of a function as per the CBSE 2025-26 syllabus?
Calculus provides a powerful method called the First Derivative Test to find a function's maximum. The steps are:
- Step 1: Find the first derivative of the function, denoted as f'(x).
- Step 2: Solve for the critical points by setting the derivative equal to zero (f'(x) = 0) and finding where it is undefined.
- Step 3: Analyse the sign of f'(x) around each critical point. If the sign of f'(x) changes from positive to negative as you move across the critical point, that point corresponds to a local maximum.
4. What is the significance of the Second Derivative Test in identifying a maximum?
The Second Derivative Test is an efficient method to confirm if a critical point is a maximum. After finding a critical point 'c' where f'(c) = 0, you find the second derivative, f''(x). If f''(c) < 0 (the value is negative), it confirms that the function is concave down at that point, and therefore, x = c is a point of local maximum. It often provides a quicker verification than the First Derivative Test.
5. What are some practical, real-world applications of finding the maximum of a function?
The concept of finding a maximum is crucial in many fields to achieve optimal outcomes. For instance:
- In Business and Economics, it is used to determine the production level that results in maximum profit or revenue.
- In Engineering, it helps in designing structures like beams for maximum strength or creating containers with maximum volume using a minimal amount of material.
- In Physics, it is used to calculate the maximum height a projectile can reach or the maximum displacement in an oscillating system.
6. If a function's derivative is zero at a point, does it guarantee a maximum or minimum at that point?
No, it does not. A point 'c' where the derivative f'(c) = 0 is called a critical point, which is a necessary condition for a local maximum or minimum, but it is not sufficient. This point could be a local maximum, a local minimum, or a point of inflection. A point of inflection is where the curve's concavity changes, but it is neither a peak nor a trough (e.g., the function y = x³ at x=0).
7. What are critical points, and why are they so important when determining a function's maximum?
Critical points are the points in a function's domain where the first derivative is either zero or undefined. They are fundamentally important because a function's local maxima (and minima) can only occur at these specific locations. By identifying all critical points, you create a complete list of candidates for potential maxima. The next step is then to test these points to determine which, if any, are the actual maximum values.
