Question

The value of \(c\) for which the function \(f(x)=\log \left(\frac{x^{2}+6}{5x}\right)\) satisfies Rolle's theorem in the interval \([2,3]\) is ____

• A. \(\sqrt{5}\)
• B. \(\sqrt{7}\)
• C. \(\sqrt{8}\)
• D. \(\sqrt{6}\)

Hint:

Set the derivative of f equal to zero and pick the root lying in the interval from 2 to 3.

Answer 1

The Correct Option is D

Step 1: Derivative formula
$f'(x)=\frac{x^{2}-6}{x(x^{2}+6)}$ after combining the two fractions.

Step 2: Set to zero
$f'(c)=0$ gives $c^{2}-6=0$.

Step 3: Choose the root
$c=\pm\sqrt{6}$, and only $\sqrt{6}\approx 2.45$ lies in $(2,3)$.

Final Answer:
The required value is $\sqrt{6}$. \[ \boxed{\text{(D) }\sqrt{6}} \]