Notes on Application of Derivative

Introduction

• Welcome by Shashank Singh Rajput
• Topic: Application of Derivative (AOD)
• Importance: One of the most crucial chapters in mathematics, especially for competitive exams like JEE.

Syllabus Overview

• Key Topics in AOD:
  • Rate of change of quantities
  • Increasing and decreasing functions
  • Maxima and minima
  • First-order derivative testing
  • Geometric representation
  • Second-order derivative testing
  • Tangent and normal
• Note: Many theorems (e.g., Rolle's theorem, Mean Value theorem) are excluded from the syllabus.

Monotonicity of Functions

Definitions:

• Increasing Function: If a function increases and then becomes constant.
• Strictly Increasing Function: If a function continuously increases without becoming constant.
• Decreasing Function: If a function decreases and then becomes constant.
• Strictly Decreasing Function: If a function continuously decreases without becoming constant.
• Neither Increasing nor Decreasing: If a function increases and decreases alternately.

Monotonic Function

• Defined as either an increasing or decreasing function.

Non-Monotonic Function

• If a function neither increases nor decreases, it is termed non-monotonic.

Finding Monotonicity at a Point

Steps:

1. Identify three points: a smaller point, a point at a, and a larger point.
2. Determine the function values at these points to check monotonicity.

Maxima and Minima

First Order Derivative Testing

1. Find critical points where f'(x) = 0 or f'(x) does not exist.
2. Determine the nature of each critical point using the sign of f'(x) before and after the point.

Second Order Derivative Testing

1. If f''(x) < 0 at a critical point, it is a point of local maxima.
2. If f''(x) > 0 at a critical point, it is a point of local minima.

Global Maxima and Minima

Procedure:

1. Find all critical points within the defined interval.
2. Calculate the function values at critical points and endpoints of the interval.
3. Identify the greatest and least values to determine global maxima and minima.

Important Techniques

• For expressions like A sin(x) ± B cos(x):
  • Maximum value: √(A² + B²)
  • Minimum value: -√(A² + B²)
• For parabolas defined by Ax² + Bx + C:
  • If A > 0, minimum value is at -D/4A
  • If A < 0, maximum value is at -D/4A

Example Problems

1. Finding the maximum value of a function within a given interval.
2. Identifying local maxima and minima using both first-order and second-order derivative tests.

Conclusion

• Importance of understanding the concepts of increasing/decreasing functions and maxima/minima.
• The role of derivatives in determining the behavior of functions.