Increasing And Decreasing Functions In Calculus, Illustration, Important Topics For JEE 2024

Increasing And Decreasing Function : Stationary point is defined as the points at which slope of tangent is equal to zero or at which curve take a U turn if slope of changes from negative to positive or positive to negative.

Change of slope form negative to positive makes the stationary point as local minima and change of slope from positive to negative makes the point as local maxima this could be shown as below graph.

In above graph as we could see has stationary point as local minima while has stationary point as local maxima.

Sometimes at stationary points curve does not takes U turn and continue to increase or decrease such as shown below

In above graph as we could see has stationary point and decreasing while has stationary point and increasing.

Critical points is defined as the points at which slope of tangent is either equal to zero known as stationary points or slope of tangent is undefined such as shown below

In above graph are critical points with slope of tangent as zero and are points with slope of tangent as undefined.

Point of Inflection

Point of inflection for a curve is defined as the points at which curve changes it concavity. It can either shift from concave upward to concave downward or concave downward to concave upward. Sign of double differential of the curve helps in identifying point of inflection if than curve is concave upward and if than curve is concave downward, change of value around point of inflection form to makes transition from concave upward to concave downward and vice versa.

In above graph has point of inflection at while has point of inflection at

As discussed, critical, stationary and point of inflection are defined with the help of derivative of the curve let’s explore it with examples.

Examples Based On Point Of Inflection And Stationary Points

Example 1: Find point of inflection for the curve ?

Sol. Double Differentiate the function with respect to x as

Equate the function to zero

Now as we know point of infection changes behaviour of slope means either increasing to decreasing or decreasing to increasing.

For , gives value of as negative and for as positive hence it’s a point of inflection and changing concavity from concave downward to concave upward.

Example 2: Find all set of stationaries points for the function and find behaviours of slope around it?

Sol. Differentiate the function with respect to as

Equate the derivative to zero

By hit and trial approach one of the factors is and remaining factor can be found using divisor method as

Behaviour of slope around is from negative to positive

Behaviour of slope around is from positive to negative

Behaviour of slope around is from negative to positive

Rapid Questions Based on point of inflection and stationary points

1. Find all set of stationaries points for the function and find behaviour of slope around it?

2. Find point of inflection for the curve ?

Advance illustrations based on point of inflection

1. Find point of inflection for the function in the interval and define change in concavity at points of inflection?

Sol. Differentiate the function with respect to as

Equate the derivative to zero

graph if cos function is shown as

Double derivative of the curve would be as equate it to zero

Graph of would be as

Form graph we could observe

value at changes from – to + when moving from to concavity from downward to upward

value at changes from + to - when moving from to concavity from upward to downward

value at changes from – to + when moving from to concavity from downward to upward

value at changes from + to – when moving from to concavity from upward to downward

value at changes from – to + when moving from to concavity from downward to upward

value at changes from + to - when moving from to concavity from upward to downward

value at changes from – to + when moving from to concavity from downward to upward

value at changes from + to - when moving from to concavity from upward to downward

Rapid Questions Based on point of inflection and stationary points

1. Find point of inflection for the function in the interval and define change in concavity at points of inflection?

2. Find point of inflection for the function in the interval if possible?