Calculus-Differential Calculus-Increasing and Decreasing Functions

Increasing and decreasing functions are calculus functions in which the value of f(x) rises and decreases with the value of x, respectively. To check the behavior of growing and decreasing functions, the derivative of the function f(x) is utilized. If the value of f(x) grows with an increase in the value of x, the function is said to be increasing, and if the value of f(x) decreases with an increase in the value of x, the function is said to be declining.

Monotonicity of a function

The monotonicity of a function can be described as the increasing or decreasing nature of a function. Monotonic functions are those functions that follow any one of these cases:

• If x and X  are two endpoints in an interval and x< X

•  If f(X) is less than or equal to f(x), then the function is a decreasing function.

• If f(X) is greater than or equal to f(x), the function is known as an increasing function.

• If f(X) is always greater than f(x), the function is known as strictly increasing.

• If f(X) is less than f(x), the function is known as strictly decreasing.

Increasing functions

Any function that is increasing at a given interval is known as an increasing function. I suppose there are two endpoints of an interval, namely, x and X, where X>x. In the case of these functions, f(x) Is less than or equal to f(X), which is quite different from a strictly increasing function. This inequality makes the function unique in the sense that some parts of this function are similar to both functions, which makes it different from strictly increasing functions. A strictly increasing function is the one where f(x) is less than f(X); it can also be written as f (x) < f (X). 

Decreasing functions

Any function which decreases in a given interval is known as a decreasing function. Now suppose that there are two endpoints in an interval, namely x and X, and f (x) is greater than or equal to f(X), then the function is a decreasing function. If however, f(x) is greater than f(X) i.e f(x) > f(X) then the function is strictly decreasing. 

Monotonic functions

• Monotonic functions are those functions that can be differentiated in a given interval of time and that are included in any one of the following categories:

• Increasing function

• Strictly increasing function

• Decreasing function

• Strictly decreasing function 

A function where df/dx = 0 is constant in a given interval.

Monotonic functions can be further explained with the help of the first derivative test, which is discussed below. 

First derivative test

Let us now understand monotonic functions with the help of the first derivative test. The derivative shows the behavior of a function at different values.

• For all the values of x, if df/dx ≤ 0, the function is a decreasing function.

• For all the values of x, if df/dx ≥ 0, the function is an increasing function.

Strict equalities can be shown as follows:

• The function will be strictly increasing if df/dx > 0.

• The function will be strictly decreasing if df/dx < 0.

Test for the increasing and decreasing functions

To test for the increasing and decreasing functions, we will use derivatives of a function. For all the four cases that we had discussed earlier (i.e., Increasing function, strictly increasing function, decreasing function, and strictly decreasing function), we have tests. However, there is something to remember before we discuss these tests. To test the monotonicity of a function f, we first calculate its derivative f′. The function should be continuous in the [a,b] interval, and it should be differentiable in (a,b). The tests are given as follows:

For a non-increasing function, the test is: f’ (x) ≤0, ∀ x ∈ (a,b)

For a non-decreasing function, the test is: f’ (x) ≥ 0, ∀ x ∈ (a,b)

For a decreasing and strictly decreasing function the test is:

f’(x) < 0 ∀ x ∈ (a,b)

For an increasing and strictly increasing function, the test would be as follows:

f’(x) > 0 ∀ x ∈ (a,b)

The monotonicity of a function can be described as the increasing or decreasing nature of a function. Suppose we have given two points x₂ < x₁, then increasing functions are the ones where f[x₁] is greater than or equal to f[x₂] and a decreasing function is ones where f[x₁] is lesser than or equal to f[x₂], a strictly increasing function is one where f[x₁] is more significant than f(x₂). A strictly decreasing function is one where f[x₁] is lesser than f[x₂]. 

Conclusion

The derivative of a function can be used to determine whether it is increasing or decreasing anywhere at any point in its domain. If f′(x) > 0, f is increasing on the interval, while f′(x) 0 indicates that f is decreasing on the interval. Or in other words

Increasing Function: A function f(x) is increasing on an interval if for any two numbers x and y in I such that x < y, we have f(x) ≤ f(y).

Decreasing Function: A function f(x) is decreasing on an interval I if for any two numbers x and y in I such that x < y, we have f(x) ≥ f(y).