Increasing And Decreasing Function, Important Topics For JEE 2024

Function : A function is identified by its mapping means function is like a machine which does not give different outputs for same input such as in for input 0 we have two outputs as hence it is not a function, graphically a function can be identified by drawing a line parallel to y axis. If drawn line intersect the curve in more than one point than the given curve is not a function. Now a function is said to be increasing if for every domain the following relation holds

Let are two random points in domain and , Now if the function is increasing then _. increasing function are categorized in two parts increasing and strictly increasing functions.

Increasing function

Increasing function : If a function is increasing than it will hold the relation as if than domain} and its graph could be as shown below

In above graph as we could see but while and

Strictly Increasing function:

Strictly Increasing function : If a function is strictly increasing than it will hold the relation as if than domain} and its graph could be as shown below

In above graph as we could see while and

Decreasing function

Decreasing function : If a function is Decreasing than it will hold the relation as if than domain} and its graph could be as shown below

In above graph as we could see but while and

Strictly decreasing function

If a function is strictly decreasing than it will hold the relation as if than domain} and its graph could be as shown below

First order derivative test (function must be differentiable) helps in identifying increasing and decreasing functions such as

1. Differentiate the function

2. if function is increasing and if function is strictly increasing.

3. if function is decreasing and if function is strictly decreasing.

Strictly increasing and strictly decreasing function may have condition as and respectively if function have slope equal to zero in discrete points. Such as shown in the graph below

In above graph at point a slope of tangent is equal to zero and it is a discrete means we don’t have two continuous points for which slope of tangent is equal to zero similarly strictly decreasing function could also be understood.

Increasing And Decreasing Behaviour Of Functions Examples

Example 1: Find the interval in which the following function increases and decreases?

Sol. Differentiate the function with respect to x

Equate the function to zero

Plot In number line as

Positive interval is where function is increasing.

Negative interval is where function is decreasing.

Example 2: Find value of for which the function decreases?

Sol. Differentiate the function with respect to x

If the function is decreasing than

Above inequality implies value of always must be greater than this is possible only when

Function Rapid Questions

Based on increasing and decreasing behaviour of functions

1. Find value of for which the function decreases?

2. Find the interval in which the following function increases and decreases?

Functions  Advance illustrations

1. Find the value of for which the function increases for all real value of x?

Sol. Differentiate the function with respect to x

If the function is increasing than

Above inequality holds only when

………(i)

……….(ii)

Intersection of equation (i) and (ii) is

Rapid Questions

1. Find the value of for which the function increases for all real value of x ?

2. Find the set of points for which the following function is increasing or decreasing?