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

To find the minimum value of the function \(f(x) = x^2 + \frac{250}{x}\) for \(x>0\):

Step 1: Compute the first derivative:

\[f'(x) = 2x - \frac{250}{x^2}\]

Step 2: Find critical points by setting the derivative to zero:

\[2x - \frac{250}{x^2} = 0\]

Solving for \(x\):

\[2x^3 - 250 = 0 \implies x^3 = 125 \implies x = 5\]

Step 3: Use the second derivative test to confirm a minimum:

\[f''(x) = 2 + \frac{500}{x^3}\]

Evaluate at \(x=5\):

\[f''(5) = 2 + \frac{500}{125} = 6\]

Since \(f''(5)>0\), \(x=5\) corresponds to a local minimum.

Step 4: Calculate the minimum value of the function:

\[f(5) = 5^2 + \frac{250}{5} = 25 + 50 = 75\]