Module 5 Quick Reference

Applications of Derivatives II – Calculus I

First Derivative Test

f' changes + to −

Local maximum

f' changes − to +

Local minimum

No sign change ⇒ no extremum (e.g., x³ at 0)

Second Derivative Test

f'(c) = 0, f''(c) > 0 ⇒ local min | f''(c) < 0 ⇒ local max | f''(c) = 0 ⇒ use FDT

Concave Up

f'' > 0

Concave Down

f'' < 0

Inflection point: f'' changes sign

Curve Sketching Checklist

1. Domain → 2. Intercepts → 3. Symmetry → 4. Asymptotes → 5. f' (inc/dec, extrema) → 6. f'' (concavity, inflection) → 7. Plot and sketch

Optimization Strategy

1. Draw & label → 2. Objective function → 3. Constraint to one variable → 4. Find critical points → 5. Verify max/min

Check domain endpoints! Verify with 2nd derivative or FDT.