## Introduction:

The increasing and decreasing functions in business mathematics are the objective life process understood in the form of variables and formulas. In these functions, the variables control the entire process of increasing and decreasing. It is more relative to the rate of change of behaviour in different intervals. The functions are the critical parts responsible for increasing and decreasing processes. The function increases, decreases, and even remains constant in the various intervals for the domain they belong to. The increasing and decreasing functions usually occur with a direct impact on the quality of the function.

## Explanation:

The increasing and decreasing behaviour of the function is known as the monotonic function. The rate of change of variables in a specific time with the variation in the quality of function is called the increase and decrease function. These functions are related to the derivatives and reflect the process as an increasing and decreasing system. So let’s go through the mathematical representation of the increase and decrease system.

## Increasing functions:

The derivatives are included in the mathematical expressions to understand the increasing functions in calculus with their variables. Suppose y = f (a) is the differential function, a constant point in all the domain systems. The interval in which the function occurs is x=a, b. In the increasing system, any two variables such as a1 and a2 are present in an interval a, which is represented in the forms of a1 < a2 this condition, a situation of inequality reflects with the functions.

The function derives its value as f (a1) ≤ f (a2). It is a condition where the f (a) is the monotonic increasing function of the variables. The condition can also be under with strictly increasing state of the variables.

## Decreasing functions:

The decreasing function in calculus states its value with variables a1 and a2 . The derivative is in the form of these variables and represented with values in a1 > a2 . The function changes its value with this variation and becomes f (a1) ≥ f (a2). This condition is the decreasing function if the variable f (a) is within the interval.

The decreasing function is also monotonic. The condition of the function here is f (a), which acts relatively opposite to the decreasing functions. There is the further stage where the variables reach the strictly decreasing functions.

## Monotonic function:

The definition states that any variable that can strictly increase or decrease with the extreme state without changing the sign of its first derivative is the monotonic function. It relates to the meaning of being in a single state of function, which is monotone. If the variable follows the increasing or decreasing function in a single state is the monotonic function. The variable is tested for its monotonic function by the specific test.

## First derivative test:

The first derivative test uses the variables to behave in a fixed time interval to notify its functioning. This test determines the state of the variable whether they fall in the monotonic category or not.

If all the variables involved increase simultaneously, it is an increasing function. If the behaviour is adverse, they belong to the decreasing function in calculus with monotonic nature. In the monotonic state the first derivative of the variables does not change their sign. Here is the mathematical representation:

- If df/da≥0 for every an (x, y), then f (a) is the increasing function in that particular domain(x, y).
- When the df/da≤0 in all an (x, y), this condition of the function f (a) is the decreasing function in the domain(x, y).
- Strictly increasing function= df/dx>0
- Strictly decreasing function = df/dx<0

## Properties of monotonic functions:

The decreasing and increasing function in calculus with monotonic nature have specific algebraic properties which involve the nature of the variable with their rate of change of value in the particular time interval. These properties are essential in finding the nature of the function with mathematical derivatives.

Here are the mentions:

- If the function of ‘a’ and ‘b’ variables is increasing (decreasing) in a particular interval (x, y), then their sum of functions will also increase and decrease.
- When the function f is increasing (decreasing) in the interval (x, y), the opposite function (–f) will decrease (increase).
- The function of ‘f’ when increases (decreases) in the time (x, y) the inverse function 1/f is opposite i.e. decreasing (increasing).
- When the function of ‘a’ and ‘b’ are increasing in the condition when a≥0 and b≥0, then both variables’ product also increases in a similar domain.
- When function ‘a’ is increasing on interval (x, y) and function b is increasing on interval (p, q), with a: (x, y) – (p, q), then composite function forms in terms of ‘a’ and ‘b’. The function also increases within their interval and domain.

### Conclusion:

The monotonic increasing and decreasing function in the calculus represents the behaviour of the variable in a particular interval. This interval and derivatives change in different domains and increase and decrease according to their nature. All the values define their state according to the time they are present and the behaviour of the variables. The monotonic properties keep all the variables in a similar state of function. The function has the variable which changes with the rate of change of quality with the increasing and decreasing function.