Question

The minimum value of \(\left(x^2 + \frac{250}{x}\right)\) is:

• A. 25
• B. 50
• C. 75
• D. 85

Hint:

To find the minimum or maximum of a function, use the first derivative test by setting it to zero and then confirm by checking the second derivative.

Answer 1

The Correct Option is C

Given the function \(f(x) = x^2 + \frac{250}{x}\), where \(x>0\), we want to find its minimum value. Step 1: Find the derivative of \(f(x)\) with respect to \(x\): \[ f'(x) = \frac{d}{dx}\left(x^2 + \frac{250}{x}\right) = 2x - \frac{250}{x^2} \] Step 2: Set the derivative to zero to find critical points: \[ 2x - \frac{250}{x^2} = 0 \] Multiply both sides by \(x^2\): \[ 2x^3 - 250 = 0 \implies 2x^3 = 250 \implies x^3 = 125 \] \[ x = \sqrt[3]{125} = 5 \] Step 3: Check the second derivative to confirm minimum: \[ f''(x) = \frac{d}{dx}\left(2x - \frac{250}{x^2}\right) = 2 + \frac{500}{x^3} \] At \(x=5\), \[ f''(5) = 2 + \frac{500}{125} = 2 + 4 = 6>0 \] Since \(f''(5)>0\), \(x=5\) is a point of local minimum. Step 4: Calculate the minimum value: \[ f(5) = 5^2 + \frac{250}{5} = 25 + 50 = 75 \]