find two positive numbers whose product is 38 and whose sum is a minimum. the two numbers are . (type an…

Explanation:

Step1: Let the two numbers be x and y

Let (xy = 38), so (y=\frac{38}{x}). The sum (S=x + y=x+\frac{38}{x}), where (x>0).

Step2: Find the derivative of S

Differentiate (S(x)=x+\frac{38}{x}=x + 38x^{-1}) with respect to (x). Using the power - rule ((x^n)^\prime=nx^{n - 1}), we get (S^\prime(x)=1-38x^{-2}=1-\frac{38}{x^{2}}).

Step3: Set the derivative equal to 0

Set (S^\prime(x)=0), then (1-\frac{38}{x^{2}} = 0). Add (\frac{38}{x^{2}}) to both sides: (1=\frac{38}{x^{2}}). Cross - multiply to get (x^{2}=38). Since (x>0), (x=\sqrt{38}).

Step4: Find the value of y

Since (y = \frac{38}{x}) and (x=\sqrt{38}), then (y=\frac{38}{\sqrt{38}}=\sqrt{38}).

Answer:

(\sqrt{38},\sqrt{38})