Second derivatives, stationary points, optimisation and related rates

Learning objectives

Find the second derivative, use it to describe concavity, and relate the graphs of f, f′ and f″ to one another

Locate local maximum and minimum points and test their nature using the sign of the first derivative or the second derivative

Find points of inflexion with zero and non-zero gradient using the change of sign of f″(x), and sketch curves showing all key features

Solve optimisation problems by forming a function from a context with constraints, finding its stationary points and justifying that the value found is a maximum or a minimum

(HL) Solve related rates of change problems by differentiating a relationship between variables with respect to time using the chain rule

(HL) Tackle harder optimisation problems, including those involving implicit relationships, trigonometric functions or more than one constraint