Question

\(\mathrm{f}(x)=\frac{x}{2}+\frac{2}{x}, x \neq 0\) is strictly decreasing in

• A. \((2,3)\)
• B. \((1,3)\)
• C. \((-2,2)\)
• D. \((1,2)\)

Hint:

Differentiate and find where the derivative is negative, remembering that x = 0 is excluded.

Answer 1

The Correct Option is D

Step 1: Critical points
$f'(x)=0$ at $x=\pm2$.

Step 2: Test a point
At $x=1.5$: $f'=\frac12-\frac{2}{2.25}<0$, so $f$ is decreasing there.

Step 3: Test the options
Options (A) and (B) include values greater than $2$ where $f'>0$, and (C) includes $0$. Only (D) works.

Final Answer:
Option (D), $(1,2)$, is the interval of decrease. \[ \boxed{\text{(D) }(1,2)} \]