Graphs & max/min

V1 — Candidates where slope vanishes.

Builds on: f′ as a function (intro + later rule tools).

Tool gained: critical points — solve f′(x) = 0 (and note where f′ undefined in the domain).

1. Principle. At a smooth local peak or valley the tangent is flat — so f′ = 0 is the first place to look. Corners/cusps can also be candidates where f′ fails.
2. Model. Find x in the domain with f′(x) = 0 or f′ undefined; these are critical numbers.
3. Prediction / Answer. f(x) = x³ − 3x → f′ = 3x² − 3 = 0 at x = ±1. Those two x-values are the candidates for local max/min.

V2 — Classify with sign of f′ (or f″).

Builds on: critical points (V1) + ability to evaluate/sign f′.

Tool gained: first-derivative sign chart (and optional second-derivative test) to label local max/min.

1. Principle. If slope flips +→−, you rode over a peak; −→+ is a valley. Alternatively f″(c) < 0 at a critical point suggests a local max, > 0 a local min (when f″ exists and ≠ 0).
2. Model. Sign chart of f′ on intervals around critical points; or test f″(c).
3. Prediction / Answer. For f(x) = x³ − 3x: f″(x) = 6x → f″(1) = 6 > 0 local min; f″(−1) = −6 < 0 local max. Values f(−1)=2, f(1)=−2.

V3 — Closed interval / job: pick the best value.

Builds on: classify critical points (V2) — compare also endpoints when the domain is closed.

Tool gained: extreme-value check — candidates + endpoints → choose global max/min on a closed interval (or state open-domain limits).

1. Principle. A real job often has a allowed window of x. The best and worst values occur at critical points inside or at the ends of that window.
2. Model. On [a,b]: evaluate f at critical points in (a,b) and at a,b; pick largest/smallest.
3. Prediction / Answer. Fence length 40 m for a rectangular pen against a wall: maximize area A(x). Critical point from A′=0 plus domain ends — classic transfer of the same tool.