find two positive numbers whose product is 187 and whose sum is a minimum.\nthe two numbers are\n(type an…

Explanation:

Step1: Define variables

Let the two positive numbers be (x) and (y). Given (xy = 187), so (y=\frac{187}{x}). The sum (S=x + y=x+\frac{187}{x}), (x>0).

Step2: Find the derivative of the sum function

Differentiate (S(x)) with respect to (x). Using the power rule ((x^n)^\prime=nx^{n - 1}), (S^\prime(x)=1-\frac{187}{x^{2}}).

Step3: Find the critical points

Set (S^\prime(x) = 0), then (1-\frac{187}{x^{2}}=0). [ \begin{align*} 1&=\frac{187}{x^{2}}\ x^{2}&=187\ x&=\sqrt{187}\quad(x>0) \end{align*} ]

Step4: Check the second - derivative

Differentiate (S^\prime(x)) to get (S^{\prime\prime}(x)=\frac{374}{x^{3}}). When (x = \sqrt{187}), (S^{\prime\prime}(\sqrt{187})=\frac{374}{(\sqrt{187})^{3}}>0), so (S(x)) has a minimum at (x=\sqrt{187}). When (x=\sqrt{187}), (y=\frac{187}{\sqrt{187}}=\sqrt{187})

Answer:

(\sqrt{187},\sqrt{187})