Monotonicity is one of the important concepts of the application of derivatives. The monotonicity of a function gives an idea about the behaviour of the function. A function is said to be monotonically increasing if its graph is only increasing with increasing values of the equation. Similarly, a function is monotonically decreasing if its values are only decreasing.
Monotonicity: The most useful element taken into consideration amongst the total activities of the function is their monotonic behaviour. It tells about the increasing or decreasing behaviour of the function.
Extremum: An extremum of a function is the point where we get the maximum or minimum value of the function in some interval.
Monotonicity of a Function
What is a monotonic function? Functions are known as monotonic if they are increasing or decreasing in their entire domain.
Example: f(x) = 2x + 3, f(x) = log(x), f(x) = ex are examples of increasing function, and f(x) = -x5 and f(x) = e-x are the examples of decreasing function.
Increasing function:
If x1 < x2 and F(x1) < F(x2), then the function is known as an increasing function or strictly increasing function.
Decreasing function:
For F(x) = e(-x)
If x1 < x2 and F(x1) > F(x2), then the function is known as decreasing function or strictly decreasing function.
Non-monotonic Function
The functions which are increasing as well as decreasing in their domain are known as non-monotonic functions.
Example: f(x) = sin x, f(x) = |x| are examples of non-monotonic functions. But f(x) = sin x is increasing in [0, Π/2], or we can say it is monotonic in [0, Π/2]
Monotonicity of a Function at a Point
A function is said to be monotonically increasing at x = a, if it satisfies
f(a + h) > f(a)
f(a – h) < f(a)
Where h is a very small value
A function is said to be monotonically decreasing at x = a, if it satisfies
f(a + h) < f(a)
f(a – h) > f(a)
Where h is a very small value
Note: We can talk about the monotonicity of a function at x = a, only if x = a is in the domain of the function, and we don’t need to take continuity and differentiability into consideration.
Monotonicity in an Interval
(a). For an increasing function in some interval,
If dy/dx > 0 for all the values of x belongs to that interval, then the function is known as a monotonically increasing or strictly increasing function.
(b). For a decreasing function in some interval,
If dy/dx < 0 for all the values of x belongs to that interval, then the function is known as a monotonically decreasing or strictly decreasing function.
Note: Hence to find the interval of monotonicity for a function y = f(x), we need to find out the value of dy/dx and have to solve the inequality dy/dx > 0 0r dy/dx < 0. The solution to this inequality gives the interval of monotonicity.
Note: If dy/dx = 0 for a function y = f(x), still the function can be increasing at x = a. Consider a function f(x) = x3 which is increasing at x = 0 although dy/dx = 0. This is because f(0 + h) > f(0) and f(0 – h) < f(0).
All those points where dy/dx = 0, but the function is still increasing or decreasing, are known as points of inflection, which indicate the concavity change.
Increasing and Decreasing Function
(a). For a function y = f(x) to be monotonically increasing, dy/dx ≥ 0 for all such values of interval (a,b), and equality may hold for discrete values of x.
(b). For a function y = f(x) to be monotonically decreasing, dy/dx ≤ 0 for all such values of interval (a,b), and equality may hold for discrete values of x.
Note: By discrete points, we mean all those points where dy/dx = 0 does not form any interval.
Maxima and Minima
- Local Maxima: It is defined as the point on the curve where the function value at the point is greater than the limiting function value.
- Local Minima: It is defined as the point on the curve where the function value at the point is lesser than the limiting function value.
- Global Maxima: It is the highest value of the function among the various critical points of the function.
- Global Minima: It is the least value of the function among the various critical points of the function.
Extremum of Functions
The extremum of functions is the least and the greatest value of the function. There are three different cases for all such values, and they are as follows:
Case 1: If a function y = f(x) is the strictly increasing function in an interval [a,b], then f(a) is the least value and f(b) is the greatest value, as shown in figure-1.
Case 2: If a function y = f(x) is the strictly decreasing function in an interval [a,b], then f(a) is the greatest value and f(b) is the lowest value, as shown in figure-2.
Case 3: If a function y = f(x) is non-monotonic in the interval [a,b] and is continuous, then the greatest and the least value of the function are at those points where dy/dx = 0 or where dy/dx does not exist or at extreme values, i.e. at x = a and x = b.
Points to remember
1] A function might consist of many local maxima and local minima but only one global maximum and one global minimum.
2] The local maximum or minimum value may or may not be the global maxima or global minima.
3] The local maximum value may be less than the local minimum at some point.
4] For any given continuous function, the happening of minima and maxima is alternate.
Examples on Monotonicity and Extremum of Functions
Example 1: Prove that f(x) = x – sin(x) is an increasing function.
Solution: f(x) = x – sin(x)
⇒ dy/dx = 1 – cos(x)
dy/dx ≥ 0 as cos(x) having a value in the interval [-1,1] and dy/dx = 0 for the discrete values of x and do not form an interval; hence we can include this function in a monotonically increasing function.
Example 2: Find the interval of monotonicity for f(x) = x/(log(x)).
Solution: f(x) = x/(log(x))
⇒ dy/dx = (log(x) – 1)/(log(x))2
dy/dx > 0
log x – 1 > 0
⇒ x > e
f(x) is increasing for x> e.
dy/dx < 0
⇒ log x – 1 < 0
⇒ x < e
f(x) is decreasing for x< e.
Example 3: Find the interval of monotonicity for f(x) = f(x) = x3/3 + (5x)2/2 + 6x.
Solution: f(x) = x3/3 + (5x)2/2 + 6x
⇒ f’(x) = x2 + 25x + 6
For strictly increasing:
f’(x) > 0
⇒ x2 + 5x + 6 > 0
⇒ (x+3)(x+2)> 0
⇒ x ϵ (-3,∞) U (-2,∞)
For strictly decreasing:
f’(x) < 0
⇒ x2 + 5x + 6 < 0
⇒ x ϵ (-3,-2).
Example 4: Find the extremum of function f(x) = 3x3 – 9x in the interval [-1, 4].
Solution: f(x) = 3x3 – 9x
⇒ f’(x) = 9x2 – 9 = 9(x2 – 1)
⇒ f’(x) = 0 => 9(x2 – 1) = 0 or x = ±1
⇒ f(-1) = 6
⇒ f(1) = – 6
⇒ f(4) = 156
⇒ Greatest value = 156 and least value = – 6.
Example 5: In what interval is the function sin x – cos x increasing?
Solution:
We have,
Now, f(x) is increasing function of x, if
f'(x) > o if 0<x< 3π/4 and 7π/4 < x < 2π
f(x) increases in (0, 3π/4) ⋃ (7π/4, 2π).
Frequently Asked Questions
What do you mean by extremum of function?
The point at which the function has the greatest value or the least value is called the extremum of a function.
What do you mean by the monotonicity of a function?
The increasing or decreasing character of functions is called the monotonicity of function.
What do you mean by non-monotonic function?
Non-monotonic functions are those functions that are increasing as well as decreasing in their domain.
Give an example of an increasing function.
The function f(x) = ex is an increasing function.
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