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Chapter 11: Problem 59

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 whose sum is 16 and whose product is maximum are 8 and 8, and the maximum product is 64.

Step by step solution

01

Express the problem algebraically

Suppose the two numbers are \(x\) and \(y\). We know that \(x + y = 16\), so we can express \(y\) in terms of \(x\) as \(y = 16 - x\). Their product \(P\) is defined as \(P = x \cdot y = x \cdot (16 - x )\).
02

Compute the derivative of the product

To find the maximum value of \(P\) we can use calculus and find its derivative. The derivative of \(P\) with respect to \(x\), denoted \(\frac{dP}{dx}\), is \(\frac{dP}{dx} = 16 - 2x\).
03

Equate the derivative to zero to find critical points

To find the maximum value, we set the derivative equal to zero and solve for \(x\). This gives us \(16 - 2x = 0\) or \(x = 8\).
04

Substitute the value of x back into the product equation

Plugging \(x = 8\) back into the equation for \(P\) gives \(P = 8 \cdot (16 - 8) = 64\).
05

Verification of maximum product

To ensure that this product is the maximum, we can take the second derivative of \(P\), denoted \(\frac{d^2P}{dx^2}\), and show it's less than 0: \(\frac{d^2P}{dx^2} = -2 < 0\), which indicates a maximum value.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Use point plotting to graph \(f(x)=x^{2}\) and \(g(x)=x^{2}+2\) in the same rectangular coordinate system.

Use a graphing utility to solve \((x-1)^{2}-9=0 .\) Graph \(y=(x-1)^{2}-9\) in a \([-5,5,1]\) by \([-9,3,1]\) viewing rectangle. The equation's solutions are the graph's \(x\) -intercepts. Check by substitution in the given equation.

Will help you prepare for the material covered in the next section. Use point plotting to graph \(f(x)=x^{2}\) and \(g(x)=(x+2)^{2}\) in the same rectangular coordinate system.

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Determine whether statement "makes sense" or "does not make sense" and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.
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