Question

find positive numbers x and y satisfying the equation xy = 30 such that the sum 5x + y is as small as possible
the numbers are x = √ and y = √
(type exact answers, using radicals as needed)

Explanation:

Step1: Express \(y\) in terms of \(x\)

Given \(xy = 30\), we can solve for \(y\) as \(y=\frac{30}{x}\). Then the sum \(S=5x + y\) becomes \(S(x)=5x+\frac{30}{x}\), where \(x>0\).

Step2: Find the derivative of \(S(x)\)

Using the power rule \((x^n)^\prime=nx^{n - 1}\), the derivative of \(S(x)\) is \(S^\prime(x)=\frac{d}{dx}(5x+\frac{30}{x})=5-\frac{30}{x^{2}}\).

Step3: Set the derivative equal to zero and solve for \(x\)

Set \(S^\prime(x) = 0\), so \(5-\frac{30}{x^{2}}=0\).

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Step4: Find the second - derivative of \(S(x)\) and check for minima

The second - derivative \(S^{\prime\prime}(x)=\frac{60}{x^{3}}\). When \(x = \sqrt{6}\), \(S^{\prime\prime}(\sqrt{6})=\frac{60}{(\sqrt{6})^{3}}>0\), so \(S(x)\) has a minimum at \(x=\sqrt{6}\).

Step5: Find the value of \(y\)

Substitute \(x=\sqrt{6}\) into \(y=\frac{30}{x}\), we get \(y=\frac{30}{\sqrt{6}} = 5\sqrt{6}\).

Answer:

\(x=\sqrt{6}\) and \(y = 5\sqrt{6}\)