Assume that relative maximum and minimum values are absolute...

Assume that relative maximum and minimum values are absolute maximum and minimum values. A concert promoter produces two kinds of souvenir shirt; one kind sells for $$ 18,$ and the other for $$ 25 . The total revenue from the sale of $x$ thousand shirts at $$ 18$ each and $y$ thousand at $$ 25 each is given by $$R(x, y)=18 x+25 y$$ The company determines that the total cost, in thousands of dollars, of producing $x$ thousand of the $$ 18 shirt and $y$ thousand of the $$ 25 shirt is given by $C(x, y)=4 x^{2}-6 x y+3 y^{2}+20 x+19 y-12$ How many of each type of shirt must be produced and sold in order to maximize profit?

Assume that relative maximum and minimum values are absolute maximum and minimum values.
A concert promoter produces two kinds of souvenir shirt; one kind sells for $$ 18,$ and the other for $$ 25 . The total revenue from the sale of $x$ thousand shirts at $$ 18$ each and $y$ thousand at $$ 25 each is given by
$$R(x, y)=18 x+25 y$$
The company determines that the total cost, in thousands of dollars, of producing $x$ thousand of the $$ 18 shirt and $y$ thousand of the $$ 25 shirt is given by $C(x, y)=4 x^{2}-6 x y+3 y^{2}+20 x+19 y-12$
How many of each type of shirt must be produced and sold in order to maximize profit?

Key Concepts

Hessian Matrix

The Hessian matrix is a square matrix composed of the second-order partial derivatives of the objective function. It is used in the second derivative test to determine whether a critical point is a local maximum, local minimum, or saddle point by analyzing the definiteness of the matrix.

Critical Point Analysis

Critical point analysis is the process of finding where the first partial derivatives (the gradient) of the objective function are zero. These points are candidates for local extrema (maximums or minimums) and provide initial solutions that need to be further verified as global or local optima.

Quadratic Optimization

Quadratic optimization deals with optimizing functions that are quadratic in nature. Because revenue and cost functions in many practical problems are modeled as quadratic functions, their optimization often involves solving linear equations derived from setting their first derivatives to zero, with the additional use of the Hessian matrix to classify the extrema.

Objective Function

The objective function is the mathematical expression that represents what is to be optimized—in this case, profit, which is defined as total revenue minus total cost. It encapsulates all the factors involved in earning revenue and incurring costs and serves as the target function for maximization or minimization.

Multivariable Optimization

Multivariable optimization involves finding the maximum or minimum of functions with more than one variable. In problems that depend on several decision variables, such as production levels, techniques like setting partial derivatives to zero are used to identify potential optimal points.

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