Optimizing TV Production Allocation with Cost Minimization

Suppose you make TV sets at two different factories, Factory A and Factory B. You produce x TVs at Factory A, y TVs at Factory B, and the cost given by the production of x TVs and y TVs is the function $6x^2 + 12y^2$. You need to produce exactly 90 TV sets a month. Determine how many TVs should be produced at each factory to minimize the cost.

This problem addresses the concept of optimization in the context of multivariable functions, where the goal is to minimize the cost of producing TV sets at two different factories. The main strategic approach involves using the method of Lagrange multipliers, which is a powerful technique for finding the local maxima and minima of a function subject to equality constraints. In this scenario, the constraint is the total number of TVs that must be produced, and the function to minimize is the cost associated with producing those TVs. By introducing a Lagrange multiplier, you incorporate this constraint into the objective function, allowing you to solve the resulting equations for optimal values.

Conceptually, this problem highlights the importance of balancing production efficiently between two resources. In practical applications, such as manufacturing and economics, understanding how to optimize allocations while minimizing costs is a key skill. Moreover, the quadratic nature of the cost function in this problem exemplifies how increases in production do not necessarily lead to linear increases in cost. Instead, the squared terms indicate that cost increases more sharply with an increase in production at each factory, reinforcing the necessity to optimize.

Working through such problems enhances one's understanding of multivariable calculus and shows how it can be employed to solve real-world problems. Students learn how to leverage mathematical techniques to make informed decisions, a crucial ability in both academic and professional settings.