Question

Find two numbers whose difference is 78 and whose product is as small as possible. [Hint: Let x and x-78 be the two numbers. Their product can be described by the function f(x) =x(x - 78).]

Answer 1

Two numbers whose difference is 78 and whose product is minimized are 39 and -39, based on the quadratic function f(x) = x(x - 78).

Step-by-step explanation:

The student is asked to solve an optimization problem within the realm of algebra. Specifically, they are to find two numbers whose difference is 78 and whose product is as small as possible. By introducing a variable x to represent one of the numbers, the other number can be represented as x - 78, according to the given difference. The product of these two numbers is then given by the function f(x) = x(x - 78).

The goal is to minimize the function f(x). This can be done by finding the derivative of the function, f'(x), and determining where it is equal to zero, which indicates possible minima or maxima. Calculating f'(x) = 2x - 78 and setting it equal to zero gives us the equation 2x - 78 = 0, which when solved gives x = 39. Plugging this back into the original function gives us a product f(39) = 39(39 - 78) = -1521, which shows that the product is minimized when both numbers are 39 and -39 respectively, using the fact that the minimum occurs when x is at the vertex of the parabola described by the quadratic function.