Question

of all numbers that add to 68, find the pair that has the maximum product. that is, maximize the objective function q = xy subject to the constraint x + y = 68. the values of x and y that have the maximum product are x = \\(\square\\) and y = \\(\square\\) (type integers or simplified fractions). the maximum product of x and y is q = \\(\square\\) (type an integer or a simplified fraction).

Explanation:

Step1: Express y in terms of x

From constraint \(x + y = 68\), get \(y = 68 - x\)

Step2: Substitute into product function

\(Q = x(68 - x) = -x^2 + 68x\)

Step3: Find vertex of quadratic

Vertex x-coordinate: \(x = -\frac{b}{2a} = -\frac{68}{2(-1)} = 34\)

Step4: Calculate y and Q

\(y = 68 - 34 = 34\), \(Q = 34×34 = 1156\)

Answer:

x = 34, y = 34
Q = 1156