Applied Optimization Problems

From situation to function

Constraints and domains

Interpreting and verifying

Optimization in economics

In solving an applied optimization problem, which step comes first?

Why must the domain of an applied optimization function be checked before differentiating?

The open box from a 12 unit square sheet has volume V(x) = x(12 - 2x)^2 on [0, 6]. What cut size x maximizes the volume?

A rectangular field against a river uses 100 meters of fence on three sides, area A = x y with 2x + y = 100. What is the area as a function of x alone?

Under what conditions does an applied optimization's optimum occur at an endpoint rather than an interior critical point?

In the box problem, besides the endpoint comparison, which additional check certifies x = 2 as a maximum?

The open box volume V(x) = x(12 - 2x)^2 on [0, 6] is maximized at x = 2. What is the maximum volume? Answer with a number.

The fenced area A(x) = x(100 - 2x) has derivative 100 - 4x. At what x is the area maximized? Answer with a number.

In the box problem the domain of V(x) = x(12 - 2x)^2 is [0, 6]. What is the right endpoint of the domain? Answer with a number.

For A(x) = x(100 - 2x), the second derivative is constant. What is its value? Answer with a number.