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: Understanding the Concept
Rolle's theorem needs \(f(a)=f(b)\), and then some \(c\) in \((a,b)\) has \(f'(c)=0\).

Step 2: Check the endpoints
\(f(2)=\log\frac{10}{10}=0\) and \(f(3)=\log\frac{15}{15}=0\), so the theorem applies.

Step 3: Find c
\(f(x)=\log(x^{2}+6)-\log 5x\), so \(f'(x)=\frac{2x}{x^{2}+6}-\frac{1}{x}\).
Setting it to zero, \(2x^{2}=x^{2}+6\), so \(x^{2}=6\) and \(c=\sqrt{6}\), which lies in \([2,3]\).

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