Question

find two positive numbers that satisfy the given requirements. (enter your answers as a comma - separated list.) the product is 158 and the sum is a minimum.

Explanation:

Step1: Define variables

Let the two positive numbers be \(x\) and \(y\). Given \(xy = 158\), so \(y=\frac{158}{x}\). The sum \(S=x + y=x+\frac{158}{x}\), \(x>0\).

Step2: Find the derivative of the sum function

Differentiate \(S(x)\) with respect to \(x\). Using the power rule \((u + v)^\prime=u^\prime + v^\prime\) where \(u = x\) (\(u^\prime=1\)) and \(v=\frac{158}{x}=158x^{-1}\) (\(v^\prime=- 158x^{-2}=-\frac{158}{x^{2}}\)). Then \(S^\prime(x)=1-\frac{158}{x^{2}}\).

Step3: Find the critical points

Set \(S^\prime(x) = 0\), so \(1-\frac{158}{x^{2}}=0\).

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Step4: Check the second - derivative

Differentiate \(S^\prime(x)\) to get \(S^{\prime\prime}(x)=\frac{316}{x^{3}}\). When \(x = \sqrt{158}\), \(S^{\prime\prime}(\sqrt{158})=\frac{316}{(\sqrt{158})^{3}}>0\), so \(S(x)\) has a minimum at \(x=\sqrt{158}\).
When \(x=\sqrt{158}\), \(y=\frac{158}{\sqrt{158}}=\sqrt{158}\).

Answer:

\(\sqrt{158},\sqrt{158}\)