Maxima & Minima (One Variable) – Differential Calculus for GATE, ESE, AE/JE

Maxima and Minima are important applications of Differential Calculus. This topic is frequently asked in GATE, ESE, AE/JE exams and is highly scoring with proper concept clarity.

Basic Concept

Maxima and minima refer to the highest and lowest values of a function in a given interval.

• Local Maximum: Highest value in neighborhood
• Local Minimum: Lowest value in neighborhood

Necessary Condition

For maxima or minima at x = a:

f’(a) = 0

Such points are called critical points.

Sufficient Conditions (Second Derivative Test)

• If f’’(a) > 0 → Minimum
• If f’’(a) < 0 → Maximum
• If f’’(a) = 0 → Test fails (use higher derivative test)

First Derivative Test

• If f’ changes from + to − → Maximum
• If f’ changes from − to + → Minimum

Steps to Solve

1. Find first derivative f’(x)
2. Solve f’(x) = 0 → critical points
3. Find second derivative f’’(x)
4. Apply test to determine maxima/minima

Solved Examples

Example 1

f(x) = x²

f’(x) = 2x → 0 at x = 0
f’’(x) = 2 > 0

Minimum at x = 0

Example 2

f(x) = x³ − 3x²

f’(x) = 3x² − 6x = 3x(x − 2)

Critical points: x = 0, 2

f’’(x) = 6x − 6

• x = 0 → f’’ = −6 → Maximum
• x = 2 → f’’ = 6 → Minimum

Example 3

f(x) = x³

f’(x) = 3x² → 0 at x = 0
f’’(x) = 6x → 0

Test fails → no maxima/minima (point of inflection)

Example 4

If f’(x) = 0 and f’’(x) < 0 → ?

Maximum

Important Results

• f’(x) = 0 is necessary but not sufficient
• Second derivative gives nature of extrema
• If test fails → check higher derivatives

Shortcut Tricks

• Always check second derivative quickly
• Polynomial → easy differentiation
• Use sign change method if needed
• Avoid skipping steps

Questions

Q1. Condition for maxima?

Answer: f’ = 0, f’’ < 0

Q2. If f’’ = 0?

Answer: Test fails

Exam Tips

• Don’t stop at f’ = 0
• Always check second derivative
• Be careful with sign errors
• Practice PYQs

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