Lecture 5
USING DIFFERENTIALS IN OPTIMIZATION

1. One-variable case

y = f(x)

1.1 Differential of y

dy = f'(x)dx

1.2 First-order condition (FOC)

• f'(x) = 0
• dy = 0

1.3 Differential of dy

d(dy) = d[f'(x)dx]
d²y = [f"(x)dx]dx ... since dx is fixed
d²y = f"(x)dx²
• f" < 0: d²y < 0 [same as d(dy) < 0]
• f" > 0: d²y > 0 [same as d(dy) > 0]

1.4 Second-order condition (SOC)

Maximum: d(dy) < 0
• f" < 0

Minimum: d(dy) > 0

• f" > 0

2. Two-variable case

y = f(x₁,x₂)

2.1 Differential of y

dy = f₁dx₁ + f₂dx₂

2.2 First-order conditions

dy = 0
• f₁ = 0
• f₂ = 0

2.3 Differential of dy

d(dy) = d[f₁(x₁,x₂)dx₁ + f₂(x₁,x₂)dx₂]

d²y = df₁(x₁,x₂)dx₁ + df₂(x₁,x₂)dx₂

d²y = (f₁₁dx₁ + f₁₂dx₂)dx₁ + (f₂₁dx₁ + f₂₂dx₂)dx₂

d²y = f₁₁dx₁² + 2f₁₂dx₁dx₂ + f₁₂dx₂²

2.4 Second-order conditions

For a maximum: d²y < 0
• f₁₁ < 0
• f₂₂ < 0
• f₁₁f₁₂ > f₁₂²

For a minimum: d²y > 0

• f₁₁ > 0
• f₂₂ > 0
• f₁₁f₁₂ > f₁₂²