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4.Applications of Derivatives (Maxima–Minima)
Class 12 • Mathematics • NCERT
Applications of Derivatives – Maxima & Minima
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This lesson explains how derivatives are used to find increasing–decreasing functions and to determine maximum and minimum values, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand increasing and decreasing functions.
- Learn critical points and stationary points.
- Find maxima and minima using derivatives.
- Solve NCERT board-level problems.
1. Increasing and Decreasing Functions
A function f(x) is said to be increasing or decreasing based on the sign of its first derivative.
If f′(x) > 0 → f(x) is increasing
If f′(x) < 0 → f(x) is decreasing
If f′(x) < 0 → f(x) is decreasing
2. Critical and Stationary Points
Points where f′(x) = 0 or f′(x) is not defined are called critical points.
Points where f′(x) = 0 are called stationary points.
3. Maxima and Minima
A function attains maximum or minimum value at a point where its derivative changes sign.
f′ changes from + to − → Maximum
f′ changes from − to + → Minimum
f′ changes from − to + → Minimum
4. First Derivative Test
The first derivative test is used to find the nature of stationary points.
Check the sign of f′(x) on either side of critical point.
5. NCERT Solved Example
Find maxima or minima of f(x) = x³ − 3x² + 4
f′(x) = 3x² − 6x = 3x(x − 2)
Critical points: x = 0, 2
Sign of f′ changes from + to − at x = 0 → Maximum
Sign of f′ changes from − to + at x = 2 → Minimum
6. Important NCERT Notes
• Only first derivative test is used in NCERT
• Second derivative test is optional in boards
• Always check sign change of f′(x)
• Maxima–minima questions carry high weightage
• Second derivative test is optional in boards
• Always check sign change of f′(x)
• Maxima–minima questions carry high weightage
Practice Questions (NCERT)
- Define increasing function.
- What is a stationary point?
- State condition for maximum.
- State condition for minimum.
- Find critical points of f(x) = x² − 4x.
- At which point f′(x) = 0?
- What test is used to find maxima?
- Is second derivative test compulsory?
- Find nature of stationary point if sign changes from − to +.
- Can maxima occur when f′(x) = 0?
✅ Show Answer Key
- Function with f′(x) > 0
- Point where f′(x) = 0
- f′ changes from + to −
- f′ changes from − to +
- x = 2
- Stationary point
- First derivative test
- No
- Minimum
- Yes
© Aviate Learning – Applications of Derivatives (NCERT Class 12)