Analytical Applications

turning f and f′ into a full story about a graph 📊

🏆 extrema vocabulary

Absolute (global) max/min on an interval: largest/smallest value attained.

Local max/min: higher/lower than all nearby points. Critical points: where f′ = 0 or f′ DNE (with f defined there). Candidates for extrema: critical points and endpoints (on closed intervals).

📜 MVT & EVT

Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), then ∃ c in (a,b) with f′(c) = [f(b)−f(a)]/(b−a).

Extreme Value Theorem: continuous f on closed [a,b] attains absolute max and min on that interval.

📈 increasing / decreasing tests

If f′ > 0 on an interval, f increases there; if f′ < 0, f decreases. First derivative test: sign changes of f′ around a critical point → local max/min (or neither if no sign change).

🌊 concavity & inflection

f is concave up where f″ > 0 (tangent below graph); concave down where f″ < 0. Inflection point: where concavity changes (often f″ = 0 or DNE, but verify sign change).

Second derivative test at critical c: if f′(c)=0 and f″(c)>0 → local min; f″(c)<0 → local max; f″(c)=0 → inconclusive.

🎯 optimization

Translate words into one variable with a domain, find critical points, compare with endpoints, interpret in context (units!).

✏️ quick practice

Find critical numbers of f(x) = x³ − 3x² and classify local extrema.
Where is f(x) = x e⁻ˣ concave up?
Maximize the area of a rectangle with perimeter 40.

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integration time — the fun kind of adding ∫