Optimization problems and behavior of implicit relations

Learning objectives

Translate an applied problem into a function of a single variable to be maximized or minimized, using a constraint equation to eliminate the other variable, and state the domain of the function

Find the critical points of the objective function and use the first derivative test, the second derivative test or the candidates test to identify the absolute extremum

Justify that the value found is an absolute maximum or minimum and answer the question actually asked, with the correct units, including geometric problems about areas, volumes and distances

Determine where an implicitly defined curve has horizontal or vertical tangents, and whether it is increasing or decreasing at a point, using dy/dx

Determine the concavity of an implicitly defined curve at a point, and classify a critical point of the relation, using the second derivative found by implicit differentiation