Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
Introduction
Increasing and decreasing functions describe how the value of a function changes as the input value increases. This topic is an important part of Applications of Derivatives and helps students study graph behaviour, monotonicity, and later ideas such as maxima and minima.
Maharashtra State Board: Class 12
Definition: Increasing Function
A function ( f(x) ) is said to be an increasing function on ((a, b)) if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)
Strictly Increasing Function:
- If x₁ < x₂ ⇒ f(x₁) < f(x₂)
Maharashtra State Board: Class 12
Definition: Decreasing Function
A function ( f(x) ) is said to be a decreasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)
Strictly Decreasing Function:
- If x₁ < x₂ ⇒ f(x₁) > f(x₂)
Maharashtra State Board: Class 12
Definition: Monotonic Function
A function ( f ) is said to be monotonic in an interval if it is either increasing or decreasing in that interval.
Graphical Understanding
As x increases (moving left to right),
- if the height of the graph increases, the function is increasing.
-
If the height of the graph decreases, the function is decreasing.
 |
 |
 |
| Strictly increasing function (i) | Strictly decreasing function (ii) | Neither increasing nor decreasing function (iii) |
Example 1
Show that the function \[f\] given by
is increasing on R.
Solution: Note that
\[= 3(x^2 - 2x + 1) + 1\]
\[= 3(x - 1)^2 + 1 > 0\], in every interval of R
Therefore, the function \[f\] is increasing on R.
Example 2
Find the intervals in which the function \[f\] given by \[f(x) = 4x^3 - 6x^2 - 72x + 30\] is (a) increasing (b) decreasing.
Solution: We have
or \[f'(x) = 12x^2 - 12x - 72\]
\[= 12(x^2 - x - 6)\]
\[= 12(x - 3)(x + 2)\]
Therefore, \[f'(x) = 0\] gives \[x = -2, 3\]. The points \[x = -2\] and \[x = 3\] divide the real line into three disjoint intervals, namely, \[(-\infty, -2), (-2, 3)\] and \[(3, \infty)\].
Fig 6.4
In the intervals \[(-\infty, -2)\] and \[(3, \infty), f'(x)\] is positive while in the interval \[(-2, 3)\], \[f'(x)\] is negative.
Consequently, the function \[f\] is increasing in the intervals \[(-\infty, -2)\] and \[(3, \infty)\] while the function is decreasing in the interval \[(-2, 3)\].
However, \[f\] is neither increasing nor decreasing in R.
| Interval | Sign of f′(x) | Nature of the function f |
|---|---|---|
| (−∞, −2) | (−)(−) > 0 | f is increasing |
| (−2, 3) | (−)(+) < 0 | f is decreasing |
| (3, ∞) | (+)(+) > 0 | f is increasing |
Maharashtra State Board: Class 12
Key Points: Increasing and Decreasing Functions
- Increasing means output does not decrease as input increases.
-
Strictly increasing means output always increases.
-
Decreasing means output does not increase as input increases.
-
Monotonic means either increasing or decreasing on an interval.
-
f′(x) > 0 implies increasing, f′(x) < 0 implies decreasing, and f′(x) = 0 on an interval implies constant behaviour.
