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Chapter 3: Problem 62

Among all pairs of numbers whose sum is \(20,\) find a pair whose product is as large as possible. What is the maximum product?

Short Answer

Expert verified
The maximum product of two numbers whose sum is 20 is 100.

Step by step solution

01

Form the Function to Maximize

Let one of the numbers be \(x\). Then the other number is \(20 - x\). Form the function to maximize, which is the product of these two numbers: \(f(x) = x(20 - x)\).
02

Derivative Calculation

Calculate the derivative of this function with respect to \(x\), \(f'(x)\). Using the product rule, the derivative can be expressed as \(f'(x) = 20 - 2x\).
03

Critical Points Calculation

Determine critical points by setting the derivative equal to zero and solving for \(x\): \(0 = 20 - 2x\). Solving for \(x\) yields the critical point, \(x = 10\).
04

Testing the Critical Point

Check whether the critical point corresponds to a maximum. You can do this by examining the values of the derivative to the left and right of \(x = 10\). If \(f'(x)\) changes sign from positive to negative at \(x = 10\), then \(f(x)\) will be a maximum at this point.
05

Solve for the Maximum Product

Calculate the maximum product by substituting the value \(x = 10\) into the original function: \(f(10) = 10 \cdot (20 - 10) = 100\).

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