Why this chapter matters
Derivative tests turn symbolic expressions into decisions about increase, decrease, extrema, and shape.
What you will learn
- Find intervals of increase and decrease using first-derivative sign analysis.
- Classify local extrema and concavity with derivative tests.
- Solve optimization problems with clear constraints and interpretations.
Lessons in this chapter
- Critical points and first derivative testLocate and classify potential extrema from derivative behavior.
- Concavity and second derivative testIdentify concavity changes and possible inflection points.
- Optimization from contextWrite objective and constraint equations before differentiating. Read the full guide →
- Curve sketching with derivative evidenceCombine intercepts, asymptotes, monotonicity, and concavity into one graph story.
Study task
Design an open-top box from a rectangular sheet by cutting equal corner squares. Build and optimize the volume model.
Chapter checkpoint
For f(x) = x^3 - 3x, where are the critical points and what type are they?
f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1), so critical points are x = -1 and x = 1. At x = -1, f changes from increasing to decreasing (local max). At x = 1, f changes from decreasing to increasing (local min).