Stationary points and optimisation

Learning objectives

Determine where a function is increasing or decreasing from the sign of f′(x), and interpret f′(x) > 0, f′(x) = 0 and f′(x) < 0 on a graph

Find the stationary points of a curve by solving f′(x) = 0, and classify each as a local maximum or a local minimum using a sign diagram or the shape of the graph

Form an expression for the quantity to be optimised in an applied problem, using a constraint to write it as a function of one variable, and state a sensible domain

Solve the optimisation problem by differentiating, solving f′(x) = 0 and checking the nature of the stationary point, then answer the question actually asked with correct units

Use a graphic display calculator to locate a maximum or minimum and to confirm an algebraic result, and comment on whether the optimum is realistic in the context