find two positive numbers whose product is 21 and whose sum is a minimum. the two numbers are (type an exact…

Explanation:

Step1: Define variables

Let the two positive numbers be (x) and (y). Given (xy = 21), so (y=\frac{21}{x}). The sum (S=x + y=x+\frac{21}{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{21}{x^{2}}).

Step3: Find the critical points

Set (S^\prime(x) = 0), then (1-\frac{21}{x^{2}}=0). [ \begin{align*} 1-\frac{21}{x^{2}}&=0\ \frac{x^{2}-21}{x^{2}}&=0\ x^{2}-21&=0\ x^{2}&=21\ x&=\sqrt{21}\quad(x>0) \end{align*} ]

Step4: Check the second - derivative

Differentiate (S^\prime(x)) to get (S^{\prime\prime}(x)=\frac{42}{x^{3}}). When (x = \sqrt{21}), (S^{\prime\prime}(\sqrt{21})=\frac{42}{(\sqrt{21})^{3}}>0). So (S(x)) has a minimum at (x=\sqrt{21}). When (x=\sqrt{21}), (y=\frac{21}{\sqrt{21}}=\sqrt{21}).

Answer:

(\sqrt{21},\sqrt{21})