find positive numbers x and y satisfying the equation xy = 10 such that the sum 5x + y is let s be the given…

Explanation:

Step1: Differentiate the objective function

Given $S = 5x+\frac{10}{x}=5x + 10x^{-1}$. Using the power - rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$, we have $S^\prime=5-10x^{-2}=5-\frac{10}{x^{2}}$.

Step2: Set the derivative equal to zero

Set $S^\prime = 0$. Then $5-\frac{10}{x^{2}}=0$. Add $\frac{10}{x^{2}}$ to both sides: $5=\frac{10}{x^{2}}$. Cross - multiply to get $5x^{2}=10$. Divide both sides by 5: $x^{2}=2$. Since $x>0$, we have $x = \sqrt{2}$.

Step3: Find the value of y

Since $xy = 10$ and $x=\sqrt{2}$, then $y=\frac{10}{x}=\frac{10}{\sqrt{2}} = 5\sqrt{2}$.

Answer:

$x=\sqrt{2}$, $y = 5\sqrt{2}$