Maxima and Minima
Maxima and minima of a function are the largest and smallest value of the function respectively either within a given range or on the entire domain. Collectively they are also known as extrema of the function. The maxima and minima are the respective plurals of maximum and minimum of a function. Before understanding maxima and minima in detail, let’s understand the local maximum and minimum value of the function first.
Local Maximum and Minimum
For finding maximum and minimum of a function, first we need to choose an interval.
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Local maximum and minimum are shown in the above graph. Let’s understand one by one in detail.
Local Maximum:
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In the above graph f(x) is a function. Interval is shown around the point a. We can say that a local maximum is a point where the height of the function at point "a" is greater than (or equal to) the height anywhere else in that interval.
Briefly, we can write
f(a) ≥ f(x) for all x in the interval.
In other words, we can say that there is no height greater than f(a).
Always keep in mind that a should be inside the interval, not at one end or the other.
Local Minimum:
It is similar to a local minimum, the only difference here is function value at point “a” in the interval is less than (or equal to) the height anywhere in that interval.
Briefly, we can write:
f(a) ≤ f(x) for all x in the interval.
Global (or Absolute) Maximum and Minimum:
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The maximum or minimum over the entire domain of the function is called an "Absolute" or "Global" maximum or minimum.
There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum of a function.
Finding Maxima and Minima Using Derivatives
Maxima and minima are found by using the concept of derivatives. As we know the concept of derivatives gives us the information regarding the gradient or slope of the function, we locate the points where the gradient is zero and these points are called turning points or stationary points. These are points associated with the largest or smallest values (locally) of the function.
How to Calculate Maxima and Minima Points?
Let’s understand this with an example.
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From the above figure, we can see before the slope becomes zero it is negative after it gets zero and again it becomes positive. It can be said dy/dx is -ve before stationary point dy/dx is +ve after a stationary point. Hence it can be said d2y/dx2 is positive at the stationary point shown. Therefore we can say that wherever the double derivative is positive it is the point of minima. Vice versa we can also say wherever the double derivative is negative it is the point of maxima on the curve. This is also called the second derivative test.
Derivative Tests:
To find the maxima and minima of any function we use the derivative test. Generally, first-order derivative and second-order derivative tests are used. Let us have a look in detail.
First Order Derivative Test:
Consider f be the function defined in an open interval I. Also, f be continuous at critical point c in I such that f’(c) = 0.
If f’(x) changes sign from positive to negative as x increases through point c, then c is the point of local maxima, and f(c) is the maximum value.
If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima, and f(c) is the minimum value.
If f’(x) doesn’t change sign as x increases through c, then c is neither a point of local minima nor a point of local maxima.
Second Derivative Test:
Consider f be the function defined on an interval I and it is twice differentiable at c.
If x = c will be the point of local maxima if f'(c) = 0 and f”(c)<0. Then f(c) will be having local maximum value.
If x = c will be the point of local minima if f'(c = 0 and f”(c) < 0. Then f(c) will be having local minimum value.
When both f'(c) = 0 and f”(c) = 0 the test fails. And that first derivative test will give us the value of local maxima and minima.
Properties of maxima and minima are as follow :
If f(x) is a continuous function in its domain, then at least one maximum or one minimum should lie between equal values of f(x).
Maxima and minima occur alternately. i.e between two maxima there is one minima and vice versa.
If f(x) tends to infinity as x tends to a or b and f’(x) = 0 only for one value x i.e.c between a and b, then f(c) is the minimum and the least value. If f(x) tends to – ∞ as x tends to a or b, then f(c) is the maximum and the highest value.
How to Find Maxima Functions?
To find the maxima of a function, we need to find a derivative of a function f(x) and find the critical numbers. Then, find the second derivative of a function f(x) and put the critical numbers. If the value is negative, the function has relative maxima at that point, if the value is positive, the function has relative maxima at that point.
FAQs on Maxima and Minima of Functions
1. What are the maxima and minima of a function?
The maxima and minima of a function refer to its largest and smallest values, respectively, within a specific interval or across its entire domain. Collectively, these points are known as extrema. A local maximum is a point where the function's value is greater than or equal to the values at its immediate neighboring points. Conversely, a local minimum is a point where the function's value is less than or equal to the values at its neighboring points.
2. How do you find the maxima and minima of a function using differentiation?
To find the maxima and minima of a function f(x) using differentiation, you follow these general steps:
- 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. The solutions are the critical points of the function. These are the potential locations for maxima or minima.
- Step 3: Use either the First Derivative Test or the Second Derivative Test to determine if each critical point corresponds to a local maximum, a local minimum, or neither.
3. What is the First Derivative Test for finding local maxima and minima?
The First Derivative Test examines the sign of the first derivative, f'(x), on either side of a critical point 'c'.
- If f'(x) changes sign from positive to negative as x passes through 'c', then f(c) is a local maximum.
- If f'(x) changes sign from negative to positive as x passes through 'c', then f(c) is a local minimum.
- If f'(x) does not change sign as x passes through 'c', then the point is neither a maximum nor a minimum (it is a point of inflection).
4. How does the Second Derivative Test for local maxima and minima work?
The Second Derivative Test is often a quicker method for classifying critical points. For a critical point 'c' where f'(c) = 0:
- If the second derivative f''(c) is less than 0 (negative), the function is concave down at that point, and f(c) is a local maximum.
- If the second derivative f''(c) is greater than 0 (positive), the function is concave up at that point, and f(c) is a local minimum.
- If the second derivative f''(c) is equal to 0, the test is inconclusive, and you must use the First Derivative Test.
5. What are some real-world examples where finding maxima and minima is useful?
The concept of maxima and minima is widely applied in various fields to solve optimization problems. For example:
- In business, it's used to determine the production level that will maximise profit or minimise cost.
- In engineering, it helps in designing structures (like a container) to have a maximum volume for a given surface area, thus minimising material usage.
- In physics, it can be used to calculate the maximum height reached by a projectile or the minimum potential energy of a system.
6. What is the difference between a local extremum and a global (or absolute) extremum?
A local extremum (maximum or minimum) is the highest or lowest point within a certain small neighborhood or interval of the function. A function can have multiple local maxima and minima. In contrast, a global extremum (or absolute extremum) is the single highest or lowest point over the function's entire domain. There can be only one global maximum and one global minimum value.
7. What are critical points, and why are they important for finding maxima and minima?
A critical point of a function f(x) is a point 'c' in its domain where the first derivative, f'(c), is either zero or undefined. Critical points are fundamentally important because local maxima and minima can only occur at these points. By finding all the critical points, we create a complete list of candidates for the locations of extrema, which we can then test individually.
8. What happens if the Second Derivative Test fails (i.e., f''(c) = 0)?
If the Second Derivative Test fails because f''(c) = 0 at a critical point 'c', it means the test provides no information about whether the point is a maximum, minimum, or a point of inflection. In this situation, you must revert to using the First Derivative Test. By checking the sign of f'(x) to the left and right of 'c', you can definitively determine the nature of the critical point.
9. Can a point of local minimum have a greater value than a point of local maximum?
Yes, this is entirely possible. A local minimum is simply the lowest point in its immediate vicinity, while a local maximum is the highest point in its vicinity. For a function with multiple 'hills' and 'valleys', the bottom of a smaller valley (a local minimum) can easily be at a higher y-value than the top of a different, smaller hill (a local maximum). This highlights the difference between local and global extrema.
10. How do you find the absolute maximum and minimum values of a continuous function on a closed interval [a, b]?
To find the absolute extrema of a continuous function on a closed interval [a, b], you use the Closed Interval Method. The steps are:
- Find all critical points of the function that lie inside the interval (a, b).
- Evaluate the function at these critical points.
- Evaluate the function at the endpoints of the interval, i.e., find f(a) and f(b).
- Compare all the values obtained from the steps above. The largest value is the absolute maximum, and the smallest value is the absolute minimum on that interval.
