Cue Column: Key Questions

• How to find intervals of increase/decrease?
• First vs second derivative test?
• How to find global extrema?
• What is the MVT?

Notes Column: Application of derivatives translates derivative information into function behavior. The three pillars are: monotonicity, extrema, and mean value theorems.

Monotonicity: f'(x) > 0 means increasing, f'(x) < 0 means decreasing. To find intervals: (1) compute f'(x), (2) find where f'(x) = 0 or DNE, (3) make a sign chart, (4) read off intervals.

Local Extrema: First derivative test: check sign change of f' at critical points. + to - = max, - to + = min. Second derivative test: at f'(c)=0, check f''(c). Negative = max, positive = min, zero = inconclusive.

Global Extrema on [a,b]: Evaluate f at all critical points AND both endpoints. Compare values. This always works for continuous functions on closed intervals (Extreme Value Theorem).

MVT: If f is continuous on [a,b] and differentiable on (a,b), there exists c where the tangent slope equals the secant slope: f'(c) = (f(b)-f(a))/(b-a). Rolle's Theorem is the special case f(a) = f(b).

Summary: Sign chart of f' gives monotonicity and local extrema. For global extrema, always check endpoints. MVT connects average rate of change to instantaneous rate.