Monotonicity - Notes, Concept and All Important Formula

MONOTONICITY

1. INCREASING / DECREASING / STRICTLY INCREASING / STRICTLY DECREASING NATURE OF A FUNCTION AT A POINT:

I. Increasing at x = a :

If f ( s ) ≤ f ( a ) ≤ f ( t ) when ever s < a < t , where s, t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be increasing at x = a .

(1) When 'a' be left end of the interval f ( a ) ≤ f ( x ) ∀ x ∈ ( a , a + h ) ∩ D_f for some h > 0 ⇒ f is increasing at x = a .

(2) When 'a' be right end of the interval f ( x ) ≤ f ( a ) ∀ x ∈ ( a − h , a ) ∩ D_f for some h > 0 ⇒ f is increasing at x = a .

II. Strictly increasing at x = a :

If f ( s ) < f ( a ) < f ( t ) when ever s < a < t where s, t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be strictly increasing at x = a

(1) When 'a' be left end of the interval f ( a ) < f ( x ) ∀ x ∈ ( a , a + h ) ∩ D_f for some h > 0 ⇒ f is strictly increasing at x = a .

(2) When 'a' be right end of the interval f ( x ) < f ( a ) ∀ x ∈ ( a − h , a ) ∩ D_f for some h > 0 ⇒ f is strictly increasing at x = a .

III. Decreasing at x = a :

If f ( s ) ≥ f ( a ) ≥ f ( t ) when ever s < a < t , where s , t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be decreasing at x = a .

(1) When 'a' be left end of the interval f ( a ) ≥ f ( x ) ∀ x ∈ ( a , a + h ) ∩ D_f for some h > 0 ⇒ f is decreasing at x = a .

(2) When 'a' be right end of the interval f ( x ) ≥ f ( a ) ∀ x ∈ ( a − h , a ) ∩ D_f for some h > 0 ⇒ f is decreasing at x = a .

IV. Strictly decreasing at x = a :

If f ( s ) > f ( a ) > f ( t ) when ever s < a < t , where s, t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be strictly decreasing at x = a .

(1) When 'a' be left end of the interval f ( a ) > f ( x ) ∀ x ∈ ( a , a + h ) ∩ D_f for some h > 0 ⇒ f is strictly decreasing at x = a .

(2) When 'a' be right end of the interval f ( x ) > f ( a ) ∀ x ∈ ( a − h , a ) ∩ D_f for some h > 0 ⇒ f is strictly decreasing at x = a .

2. INCREASING & DECREASING NATURE OF A FUNCTION OVER AN INTERVAL:

Consider an interval I ⊆ D_f

I. Increasing Over an Interval I :

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇒ f ( x₁ ) ≤ f ( x₂ ) , then f is increasing over the interval I.

II. Decreasing Over an Interval I :

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇒ f ( x₁ ) ≥ f ( x₂ ) , then f is decreasing over the interval I

III. Strictly increasing over an Interval I :

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇔ f ( x₁ ) < f ( x₂ ) , then f is strictly increasing over the interval I

IV. Strictly decreasing over an Interval I:

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇔ f ( x₁ ) > f ( x₂ ) , then f is strictly decreasing over the interval I.

Monotonic function:

If a function is either increasing or decreasing over an interval then it is said to be monotonic function over the interval. If a function is either strictly increasing or strictly decreasing over an interval then it is said to be strictly monotonic function over the interval. Some Important points:

(i) Increasing or monotonic increasing or non decreasing has same meaning. Similarly decreasing or monotonic decreasing or non increasing has same meaning.

(ii) If a function is strictly increasing, then it is also said to be increasing function, but converse is not necessarily true.

(iii) Functions which are increasing over some interval and decreasing over another interval are known as non-monotonic functions over the union of the intervals.

(iv) A function may be monotonic in a subset but may not be monotonic in a superset.

(v) Constant function is increasing as well as decreasing over any interval. So it is called monotonic function.

For differentiable functions:

Consider an interval I ( ⊆ D_f ) that can be [ a , b ] or ( a , b ) or [ a , b ) or ( a , b ]

(1) f ′ ( x ) > 0 ∀ x ∈ I ⇒ f is strictly increasing function over the interval I.

(2) f ′ ( x ) ≥ 0 ∀ x ∈ I ⇒ f is increasing function over the interval I

(3) f ′ ( x ) ≥ 0 ∀ x ∈ I and f ′ ( x ) = 0 do not form any interval (that means f ′ ( x ) = 0 at discrete points) ⇒ f is strictly increasing function over the interval I.

(4) f ′ ( x ) < 0 ∀ x ∈ I ⇒ f is strictly decreasing function over the interval I .

(5) f ′ ( x ) ≤ 0 ∀ x ∈ I ⇒ f is decreasing function over the interval L

(6) f ′ ( x ) ≤ 0 ∀ x ∈ I and f ′ ( x ) = 0 do not form any interval (that means f ′ ( x ) = 0 at discrete points) ⇒ f is strictly decreasing function over the interval I.

3. ROLLE'S THEOREM :

Let f be a function that satisfies the following three hypotheses:

(a) f is continuous in the closed interval [ a , b ]

(b) f is differentiable in the open interval ( a , b )

(c) f ( a ) = f ( b )

Then there is a number c in ( a , b ) such that f ′ ( c ) = 0 .

Conclusion : If f is a differentiable function then between any two consecutive roots of f ( x ) = 0 , there is atleast one root of the equation f ′ ( x ) = 0

4. LAGRANGE'S MEAN VALUE THEOREM (LMVT):

Let f be a function that satisfies the following hypotheses:

(i) f is continuous in a closed interval [ a , b ]

(ii) f is differentiable in the open interval ( a , b ) .

Then there is a number c in ( a , b ) such that \(f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}\)

(a) Geometrical Interpretation: Geometrically, the Mean Value Theorem says that somewhere between A and B the curve has at least one tangent parallel to chord AB.

(b) Physical Interpretations: If we think of the number ( f ( b ) − f ( a ) ) / ( b − a ) as the average change in f over [ a , b ] and f ′ ( c ) as an instantaneous change, then the Mean Value Theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.

5. SPECIAL NOTE :

One can make use of Monotonicity in identifying the number of roots of the equation in a given interval. Suppose a and b are two real numbers such that

(a) f ( x ) & its first derivative f ′ ( x ) are continuous for a ≤ x ≤ b .

(b) f ( a ) and f ( b ) have opposite signs.

(c) f ′ ( x ) is different from zero for all values of x between a & b . Then there is one & only one root of the equation f ( x ) = 0 in ( a , b ) .

Mathematics-Important Notes, concept & Formula

FAQs

When we called the function is increasing at the point ?

If f ( s ) ≤ f ( a ) ≤ f ( t ) when ever s < a < t , where s, t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be increasing at x = a .

When we called the function is strictly increasing at the point ?

If f ( s ) < f ( a ) < f ( t ) when ever s < a < t where s, t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be strictly increasing at x = a

When we called the function is decreasing at the point ?

If f ( s ) ≥ f ( a ) ≥ f ( t ) when ever s < a < t , where s , t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be decreasing at x = a .

When we called the function is strictly decreasing at the point ?

If f ( s ) > f ( a ) > f ( t ) when ever s < a < t , where s, t ∈ ( a − h , a + h ) ∩ D_f for some h > 0 , then f is said to be strictly decreasing at x = a .

When we define the function is increasing over an interval ?

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇒ f ( x₁ ) ≤ f ( x₂ ) , then f is increasing over the interval I.

When we define the function is strictly increasing over an interval ?

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇔ f ( x₁ ) < f ( x₂ ) , then f is strictly increasing over the interval I

When we define the function is decreasing over an interval ?

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇒ f ( x₁ ) ≥ f ( x₂ ) , then f is decreasing over the interval I

When we define the function is strictly decreasing over an interval ?

If ∀ x₁ , x₂ ∈ I , x₁ < x₂ ⇔ f ( x₁ ) > f ( x₂ ) , then f is strictly decreasing over the interval I.

What is monotonic function ?

If a function is either increasing or decreasing over an interval then it is said to be monotonic function over the interval.

Why constant function is called monotonic function?

Constant function is increasing as well as decreasing over any interval. So it is called monotonic function.

How do you get to know that differentiable function is increasing or strictly increasing over an interval?

Consider an interval I ( ⊆ D _f ) that can be [ a , b ] or ( a , b ) or [ a , b ) or ( a , b ] 

 (1) f ′ ( x ) > 0 ∀ x ∈ I ⇒ f is strictly increasing function over the interval I. 

 (2) f ′ ( x ) ≥ 0 ∀ x ∈ I ⇒ f is increasing function over the interval I 

 (3) f ′ ( x ) ≥ 0 ∀ x ∈ I and f ′ ( x ) = 0 do not form any interval (that means f ′ ( x ) = 0 at discrete points) 

 ⇒ f is strictly increasing function over the interval I.

How do you get to know that differentiable function is decreasing or strictly decreasing over an interval?

Consider an interval I ( ⊆ D _f ) that can be [ a , b ] or ( a , b ) or [ a , b ) or ( a , b ] 

(1) f ′ ( x ) < 0 ∀ x ∈ I ⇒ f is strictly decreasing function over the interval I . 
(2) f ′ ( x ) ≤ 0 ∀ x ∈ I ⇒ f is decreasing function over the interval L
(3) f ′ ( x ) ≤ 0 ∀ x ∈ I and f ′ ( x ) = 0 do not form any interval (that means f ′ ( x ) = 0 at discrete points) ⇒ f is strictly decreasing function over the interval I.

What Rolle's Theorem tell us about?

If f is a differentiable function then between any two consecutive roots of f ( x ) = 0 , there is atleast one root of the equation f ′ ( x ) = 0

What are the condition for Rolle's Theorem?

Let f be a function that satisfies the following three hypotheses:

(a) f is continuous in the closed interval [ a , b ]

(b) f is differentiable in the open interval ( a , b )

(c) f ( a ) = f ( b ) 

 Then there is a number c in ( a , b ) such that f ′ ( c ) = 0 .

What is the Geometrical Interpretation of Lagrange's Mean Value Theorem (LMVT) ?

Geometrically, the Mean Value Theorem says that somewhere between A and B the curve has at least one tangent parallel to chord AB.

What is the Physical Interpretation of Lagrange's Mean Value Theorem (LMVT) ?

If we think of the number ( f ( b ) − f ( a ) ) / ( b − a ) as the average change in f over [ a , b ] and f ′ ( c ) as an instantaneous change, then the Mean Value Theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.

What are the condition or hypotheses for Lagrange's Mean Value Theorem (LMVT)?

Let f be a function that satisfies the following hypotheses: 

(i) f is continuous in a closed interval [ a , b ] 
(ii) f is differentiable in the open interval ( a , b ) . 
Then there is a number c in ( a , b ) such that f ′ ( c ) = ( f ( b ) − f ( a ) ) / ( b − a ).