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
Maxima and minima are used to find the highest or lowest value of a quantity, such as greatest profit, shortest distance, maximum area, or minimum cost.
This topic forms an important part of Applications of Derivatives and connects graph interpretation, increasing-decreasing behavior, and optimization problems.
Definition: Maxima and Minima
A function may attain a maximum value at a point if its value there is greater than nearby values, and it may attain a minimum value if its value there is smaller than nearby values. These are often called extreme values.
Definition: Critical Point
A point in the domain of a function is called a critical point if either the derivative is zero there or the derivative does not exist there. Critical points are checked while locating possible maxima or minima.
Types of Extrema
| Type | Meaning |
|---|---|
| Local maximum | Greatest value in a small neighborhood of a point |
| Local minimum | Smallest value in a small neighborhood of a point |
| Absolute maximum | Greatest value on the entire given interval |
| Absolute minimum | Smallest value on the entire given interval |
Theorem: First Derivative Test
Let c be a critical point of a continuous function f:
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Local Maxima: f'(x) changes sign from positive to negative as x increases through c.
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Local Minima: f'(x) changes sign from negative to positive as x increases through c.
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Point of Inflection: f'(x) does not change sign as x passes through c (it is neither a maxima nor a minima).
Theorem: Second Derivative Test
Assume f'(c) = 0 and the second derivative exists at c:
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Local Maxima: f''(c) < 0
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Local Minima: f''(c) > 0
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Test Fails: f''(c) = 0. If this happens, you must go back and use the First Derivative Test to check if it is a maxima, minima, or point of inflection.
Absolute Maxima and Minima
For a continuous function on a closed interval \([a,b]\), absolute maxima and minima are found by comparing function values at:
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critical points inside the interval,
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the left endpoint \(a\), and
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the right endpoint \(b\).
This comparison is necessary because the largest or smallest value may occur at an endpoint.
Maharashtra State Board: Class 12
Key Points: Maxima and Minima
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Maxima and minima are extreme values of a function.
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Critical points occur where \(f'(x)=0\) or \(f'(x)\) is not defined.
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If \(f'(x)\) changes from positive to negative, the function has a local maximum.
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If \(f'(x)\) changes from negative to positive, the function has a local minimum.
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If \(f''(c) < 0\), there is a local maximum at \(x=c\).
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If \(f''(c) > 0\), there is a local minimum at \(x=c\).
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For absolute extrema on \([a,b]\), compare values at critical points and endpoints.
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Not every critical point gives a maximum or minimum.
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The second derivative test is quick, but the first derivative test is often more reliable in detailed reasoning.
