Final answer:
To find two positive numbers where their sum and 5 times the second is 268 and the product is maximized, solve the system of equations using x + 5y = 268 and maximize xy by substituting x from the first equation into the product equation and finding the derivative.
Step-by-step explanation:
To determine two positive values such that the sum of the first number and five times the second number is 268 and whose product is a maximum, we need to use a system of equations and optimization techniques. Let's call the first number x and the second number y. According to the problem, we have two equations: x + 5y = 268 and we want to maximize the product xy.
To express the product in terms of one variable, we can rearrange the first equation to solve for x: x = 268 - 5y. Substituting this expression for x into the product xy gives us a new equation: (268 - 5y)y. To find the values of y that maximize this product, we take the derivative of the equation with respect to y, set it to zero, and solve for y. The critical value of y that maximizes the product will also give us the value of x when we substitute it back into the equation x = 268 - 5y.
After finding the critical point, we need to check if it provides a maximum value by using the second derivative test or by comparing the values of the product at the endpoints and at the critical point. Please note, however, that the data provided in the Oxtoby textbook or any other specific values or formulas were not provided in the question, so this solution is based on standard mathematical techniques.