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Chapter 3: Problem 61

Among all pairs of numbers whose sum is \(16,\) find a pair whose product is as large as possible. What is the maximum product?

Short Answer

Expert verified
The pair of numbers with maximum product is (8, 8) and the maximum product is 64.

Step by step solution

01

Formulate the equation

Let's form a pair of numbers, denoting them as \(x\) and \(y\). According to the problem, the sum of the pair is \(16\). We can formulate this relationship into the equation \(x + y = 16\).
02

Solve the equation for one variable

We can solve for \(y\) in terms of \(x\). That gives us the equation \(y = 16 - x\).
03

Formulate the product equation

The problem also asks us to find the maximum possible product of the pair of numbers. So let's now create an equation for the product of \(x\) and \(y\). Substituting \(y\) from step 2 into the product gives us \(P = x(16 - x)\), where P is the product.
04

Differentiate the product equation

Now, in order to maximise the product, differentiate the product function with respect to \(x\). This gives us \(\frac{dP}{dx} = 16 - 2x\).
05

Set the derivative to zero and solve

For maximum value, set the derivative equal to zero and solve for \(x\). This gives us \(x = 8\). Using \(y = 16 - x\), we find \(y = 8\) as well.
06

Verify maximum

Verify that this solution gives a maximum by checking the signs of the derivative for values of \(x\) just less than and just more than 8. \(\frac{dP}{dx}\) changes from positive to negative as \(x\) increases through 8, so indeed, \(x = 8, y = 8\) gives a maximum product.

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