Maxima & Minima (Two Variables) – Differential Calculus for GATE, ESE, AE/JE

Maxima and Minima of functions of two variables is an important topic in Differential Calculus. It is frequently asked in GATE, ESE, AE/JE exams and involves partial derivatives and determinant test.

Basic Concept

For a function f(x, y), maxima and minima occur at points where the function attains extreme values.

Necessary Condition

For extrema at (a, b):

∂f/∂x = 0 and ∂f/∂y = 0

Such points are called critical points.

Sufficient Condition (Second Derivative Test)

Let:

r = ∂²f/∂x²
s = ∂²f/∂x∂y
t = ∂²f/∂y²

Determinant:

D = (rt − s²)

• If D > 0 and r > 0 → Minimum
• If D > 0 and r < 0 → Maximum
• If D < 0 → Saddle Point
• If D = 0 → Test fails

Steps to Solve

1. Find ∂f/∂x and ∂f/∂y
2. Solve = 0 → critical points
3. Find second derivatives
4. Compute D = rt − s²
5. Apply conditions

Solved Examples

Example 1

f(x, y) = x² + y²

∂f/∂x = 2x = 0 → x = 0
∂f/∂y = 2y = 0 → y = 0

r = 2, s = 0, t = 2
D = 4 > 0 and r > 0

Minimum at (0,0)

Example 2

f(x, y) = x² − y²

Critical point: (0,0)

r = 2, s = 0, t = −2
D = −4 < 0

Saddle Point

Example 3

f(x, y) = x² + 2y² − 2x − 4y

∂f/∂x = 2x − 2 = 0 → x = 1
∂f/∂y = 4y − 4 = 0 → y = 1

r = 2, s = 0, t = 4
D = 8 > 0 and r > 0

Minimum at (1,1)

Example 4

If D < 0, then point is?

Saddle Point

Important Results

• D = rt − s² is key for classification
• D < 0 always gives saddle point
• D > 0 → check sign of r

Shortcut Tricks

• Memorize D = rt − s²
• Focus on sign of determinant
• Avoid calculation mistakes in second derivatives
• Practice standard forms

Questions

Q1. If D > 0 and r < 0?

Answer: Maximum

Q2. If D < 0?

Answer: Saddle point

Exam Tips

• Always compute determinant carefully
• Don’t forget sign of r
• Be careful with mixed derivative (s)
• Practice PYQs

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