find positive numbers x and y satisfying the equation xy = 15 such that the sum 3x + y is as small as…

Explanation:

Step1: Express y in terms of x

From $xy = 15$, we get $y=\frac{15}{x}$.

Step2: Define the function to minimize

Let $S(x)=3x + y=3x+\frac{15}{x}$, where $x>0$.

Step3: Find the derivative of the function

Using the power - rule, if $y = 3x+15x^{-1}$, then $S^\prime(x)=3 - 15x^{-2}=3-\frac{15}{x^{2}}$.

Step4: Set the derivative equal to zero

$3-\frac{15}{x^{2}} = 0$. Then $\frac{15}{x^{2}}=3$, which gives $x^{2}=5$. Since $x>0$, we have $x = \sqrt{5}$.

Step5: Find the value of y

Substitute $x=\sqrt{5}$ into $y=\frac{15}{x}$, we get $y=\frac{15}{\sqrt{5}} = 3\sqrt{5}$.

Answer:

$x=\sqrt{5}$ and $y = 3\sqrt{5}$