Optimization Guide
Maximize or minimize step by step: build the objective, differentiate, test critical points.
Strategy: express everything with one variable, differentiate, set f′ = 0, then use the second derivative (or endpoints) to confirm max vs min.
① Answer
② Steps
③ Why it works
Optimization is calculus doing economics: the best (or cheapest, or largest) choice sits where the rate of change is zero — a critical point. The second derivative tells you the shape there: concave down (f″ < 0) means a peak, concave up means a valley. On a closed interval you must also check the endpoints, because the extreme can hide at the boundary.