The function which is increasing at a given interval of time is called an increasing function. The function of y = f(x) involves an increasing function having two points of intervals (x1 and x2) such that x1 < x2 with the functional inequality of f(x1) ≤ f(x2), then the function can be said to be an increasing function. In this inequality, some part of the increasing function remains equal with both the functions making it differ from strictly increasing function. While strictly increasing function involves that function which does not get equal to both the axis in between the increasing process of the function at the given interval of time. The strictly increasing function for the fixed interval of time having the intervals of x1 and x2 can be stated as f(x1) < f(x2). This increasing, as well as strictly increasing functions, can be easily shown on a graph with the help of the figures shown below;
The functions which are decreasing at the given interval of time are called as decreasing functions. The function y = f(x) shows that the function has the interval points of x1 and x2 through which a function can be decreasing or strictly decreasing at the given interval of time. The function which is decreasing holds the unique inequality of f(x1) ≥ f(x2). This inequality of f(x1) ≥ f(x2) makes it unique as the line of graph remains equal for both the axis in a particular interval of time, making it different from strictly decreasing function. This strictly decreasing function can be defined as the function which strictly holds the inequality and does not have the same value for both the axis at a given interval of time and shows the function as f(x1) > f(x2). The main difference between the strictly decreasing and decreasing function is that at any interval of time the function does not remain the same in strictly decreasing type function. These decreasing types of function can be easily explained with the help of the graph as shown in the following figures;
There are some steps involved in the process of finding the intervals of increasing and decreasing function, are as follows:
The functions which are differentiable at the given interval (a, b) of time and are included in any of the four categories which are increasing function, strictly increasing function, decreasing function or strictly decreasing function are called as monotonic functions. A function which includes df/dx = 0 is constant at that given interval of time. This monotonic nature of the function can be further explained in the first derivative test.
As explained above, the nature of the function can be determined with the help of this test, as it varies with the value of the derivative of the function. The derivative of the function with respect to the independent behavior of the function shows different behaviours with the different values which are as follow;
This test is done to show the increasing and decreasing nature of the derivative of the function. In the test of the first derivative, the function which is greater than or equal to 0 is called an increasing function with respect to the given interval of time (a, b). It can be expressed as if df/dx = 0 at the intervals (a, b) is said to be increasing in nature. If the derivative of the given function is