Increasing functions are functions that increase their value with the value of variables present inside them. It is an important part of calculus mathematics as it helps check the function’s behaviour, which can be increasing or decreasing. To determine an increasing function, the derivative of the respective function is calculated. If the value of the calculated derivative is more than zero, then the function is said to be an increasing function; otherwise, if the derivative of the function is less than zero, it is said to be a decreasing function.

An increasing function could also be defined as a function whose graph moves in the upward direction as we increase the value of the function on the right side of the X-axis. Hence when plotted graphically, a function is said to be increasing only if the value of f(x) ≤ f(y), where x is the X-coordinate and y is the Y-coordinate. Also, the function can also be named as a strictly increasing function only if the value of any two numbers x and y is such that x < y and f(x) < f(y). The only difference between an increasing function and a strictly increasing function is that the value of f(x) should not be equal to the value of f(y).

For a more detailed understanding of the increasing function, carry on reading the increasing function study material where we have mentioned a complete guide on functions, properties, graphical presentation, and theorems related to the concept.

Properties of Increasing Functions

Till now, you may have a brief idea about the concept of increasing functions. Now, let’s move to the algebraic properties of increasing functions. 

• Consider the value of the functions A and B to be increasing between the interval “I”. When the sum of the functions is performed, which is A+B, the sum of the variable doesn’t change its property and remains to be an increasing function in the interval.

• Viewing the function “A” as an increasing class in the interval” I” makes the conjugate value of the function, which is “-A”, decreasing in the interval.

• Similar to the point mentioned above, if the function “A” is seen as a decreasing function in the interval “I”, the opposite function, which is “-A”, becomes an increasing function in the interval.

• Consider the function “A” as an increasing function in the interval “I”. When the value of its inverse “1/A” is examined, it becomes a decreasing function during the interval.

• Similarly, when the function “A” decreases in the interval “I”, its inverse becomes an increasing function between the intervals.

• If the functions A and B are present and considered an increasing type with an interval named “I”, then the function of the functions, which is A*B, continues to remain as an increasing function in the interval.

• An increasing function is also called a non-decreasing function.

• The first derivative of the function is used to analyse the increasing or decreasing property of the function.

Constant Function

To classify a function as either increasing or decreasing, we have to use the first derivative test on the function. By doing this, we can prove the properties by taking a derivative function. For better understanding, let’s take an example with “f” being a differentiable and continuous function between the open interval (a,b). 

• If the value of the first derivative appears to be greater than zero, i.e., f′(x) > 0 for all the values between the interval [a,b], then the function is said to be increasing.

If f′(x) > 0 for each x ∈ (a, b) then the function (f) is an increasing function in interval [a, b]

• Similarly, if the first derivative of the functions has a value that appears to be lesser than zero, then the function is called to be of decreasing type.

If f′(x) < 0 for each x ∈ (a, b) then function (f) is decreasing function in interval [a, b]

• If the value of the function comes to be zero, then the function is neither increasing nor decreasing but is called a constant function. The value of the function remains unchanged in the graph, i.e., it shows a linear straight line.

 If f′(x) = 0 for each x ∈ (a, b) then function (f) is a constant function in [a, b] 

Proof of First Derivative Test Using Mean Value Theorem

We discussed the property of the first derivative earlier in our study material notes on increasing functions. It is used to classify a function as an increasing, decreasing, or constant function. The proof for proving the first derivative test as an accurate test is carried out with the help of the Mean value theorem, which is as follows:

Consider a function with,

x1, x2 ∈ [a, b] such that x1< x2 

Therefore, 

f’(c)= [f(x2 ) – f(x1)] / x2 -x1 ………..( c is a point between x1and x2 )……..(Mean value theorem)

• Increasing Function

 fꞌ(c) ≥ 0

f’(c) ≥ [f(x2 ) – f(x1)] / x2 -x1)

[f(x2 ) – f(x1)] / x2 -x1) ≥ 0

f(x2 ) – f(x1) ≥ 0

f(x2 ) ≥  f(x1)

Hence, we proved that f(c) is an increasing function

• Decreasing Function

 fꞌ(c) ≤ 0

f’(c) ≤ [f(x2 ) – f(x1)] / x2 -x1)

[f(x2 ) – f(x1)] / x2 -x1) ≤ 0

f(x2 ) – f(x1) ≤ 0

f(x2 ) ≤  f(x1)

Hence, we proved that f(c) is a decreasing function.

• Constant Function

 fꞌ(c) = 0

f’(c) = [f(x2 ) – f(x1)] / x2 -x1)

[f(x2 ) – f(x1)] / x2 -x1) = 0

f(x2 ) – f(x1) = 0

f(x2 ) =  f(x1)

Hence, we proved that f(c) is a constant function.

Conclusion

The concept of increasing functions is very crucial to learn. It helps solve most of the mathematical concepts related to calculus and statistics. Also, the property of increasing function could be a valuable tool to simplify the problem and make its solving process more efficient. This study material notes on increasing function could help you understand the topic more quickly and efficiently.