The term “monotonic” conveys the meaning of maintaining nature or the sense of “no change”. In the context of function, we think a monotonic function as the one whose successive values are either increasing or decreasing. Besides the increasing and decreasing nature of a function, there are few additional combination of these basic formats with constant function values. Clearly, the nature of function with respect to its values is considered within certain interval in which the function is continuous. For example, let us consider the graph of sine function. As a whole, the function is not monotonic as the nature of function (in terms of increasing, decreasing and constant) is not preserved over the domain of the function, which is “R”. However, if we consider an interval, say, between “0” and “π/2”, then we find that function keeps increasing with the increasing values of independent variable. Therefore, sine function is monotonic in this interval.

Monotonic function The sine function is monotonic in certain interval. There are four characteristics corresponding to four inequalities against which a function can maintain order or nature. In an interval, the successive function values can be less than (<), less than equal to (≤), greater than (>) and greater than equal to (≥). In the nutshell : A monotonic function is defined for an interval in which function is continuous. A monotonic function maintains the order (increasing or decreasing) of successive values. There are four types of monotonic function, corresponding to four inequalities.

Increasing function The successive value of function increases as the value of the independent variable increases. In other words, the function maintains the nature of “greater than”. Mathematically, If x 1 > x 2, then f x 1 > f x 2

Increasing function The successive function value is greater than previous value. The nature of a function provides an easy technique to determine range of a continuous function, corresponding to a given domain interval. For example, if domain of a continuously increasing function, f(x), is [ x 1, x 2 ] , then the least value of the function is f x 1 and greatest value of the function is f x 2 . Hence, range of the function is : Range = [ f x 1 , f x 2 ]

Derivative and nature of function We have included determination of range, using concept of derivative for illustration purpose only. This topic needs to be covered appropriately again in the context of limits and derivatives. We shall learn subsequently that derivative of a function is defined in terms of the difference between two successive values of a function, however small is the difference. It is easy to visualize, then, that if we know the nature of derivative in a given interval, then we can determine monotonic behavior of the function as well. f ′ x = L i m x → 0 f x + h − f x h As f(x+h) > f(x) for increasing function, the derivative of function with respect to independent variable “x” is a positive number. Hence, ⇒ f ′ x > 0 The derivative of an increasing function can equal to zero at certain points (for certain values of "x"). However, derivative can not be zero over an interval. ⇒ f ′ x ≥ 0 ; equality holds only at certain points It is easy to realize that if a continuous function is increasing in a specified domain, then range of the function is a continuous interval between values of function at lower and upper ends of the domain. Clearly, this reasoning will also be applicable, in case function is decreasing.

Examples

Problem 1: Determine monotonic nature of the function given by : y = x 2 Solution : The term “ x 2 ” is a non-negative number for all real values of “x”. Further consecutive value of the function is greater than previous value. Hence, function is an increasing function for all values of “x”.

Problem 2: Determine range of the function given by : f x = 2 sin π 2 4 − x 2 Solution : The expression within square root is non-negative number. Hence, ⇒ π 2 4 − x 2 ≥ 0 ⇒ x 2 − π 2 4 ≤ 0 ⇒ x 2 ≤ π 2 4 ⇒ − π 2 ≤ x ≤ π 2 Let z = π 2 / 4 − x 2 The interval of “z”, corresponding to the values of “x” is [0, π/2]. Recall plot of sine function. The "sin z" is an increasing function in the interval [0, π/2]. The least value of the function is : ⇒ f 0 = 2 sin 0 ⇒ f 0 = 0 The greatest value of function is : ⇒ f π 2 = 2 sin π 2 ⇒ f π 2 = 2 X 1 = 2 Hence, range of the function is : Range = [ 0,2 ]

Non-decreasing function The nature of function is determined against inequality “greater than equal to”. This condition of successive function value also includes constant function values. Mathematically, If x 1 > x 2 then f x 1 ≥ f x 2

Not decreasing function The successive function value is greater than or equal to previous value. In terms of derivative, a function is non – decreasing in an interval, if f ′ x ≥ 0 ; equality holds over an interval

Decreasing function The nature of function is determined against inequality “less than”. Mathematically, If x 1 > x 2 then f x 1 < f x 2

Decreasing function The successive function value is less than previous value. In terms of derivative, a function is a decreasing in an interval, if f ′ x ≤ 0 ; equality holds only at certain points

Non-increasing function The nature of function is determined against inequality “less than equal to”. This condition of successive function values also includes constant function values. Mathematically, If x 1 > x 2 then f x 1 ≤ f x 2

Non-increasing function The successive function value is less than or equal to previous value. In terms of derivative, a function is non – decreasing in an interval, if f ′ x ≤ 0 ; equality holds over an interval

Example Problem 3: Determine the nature of given function (increasing or decreasing) : f x = 2 x 3 + 3 x 2 − 12 x + 1 Solution : For illustration purpose, we shall apply the concept of derivative to find the nature of function in different intervals. The first derivative of the given polynomial function is : ⇒ f ′ x = 2 X 3 x 2 + 3 X 2 x − 12 = 6 x 2 + 6 x − 12 Clearly, the derivative is a quadratic function. We can determine the sign of the quadratic expression, using sign scheme for quadratic expression. Now, the roots of the corresponding quadratic equation when equated to zero is obtained as : ⇒ 6 x 2 + 6 x − 12 = 0 ⇒ x 2 + x − 2 = 0 ⇒ x 2 + 2 x − x − 2 = 0 ⇒ x x + 2 − 1 x + 2 = 0 ⇒ x − 1 x + 2 = 0 ⇒ x = 1, - 2 Here, coefficient of “ x 2 ” is positive. Hence, sign of the middle interval is negative and side intervals are positive. Hence,

Intervals Increasing and decreasing intervals Increasing interval = - ∞ , - 2 ∪ 1, ∞ Decreasing interval = - 2,1