MHT CET · Maths · Application of Derivatives

The value of C for which Mean value Theorem holds for the function \(\mathrm{f}(x)=\log _{\mathrm{e}} x\) on the interval \([1,3]\) is

A \(\log _3 \mathrm{e}\)
B \(\log _{\mathrm{e}} 3\)
C \(\frac{1}{2} \log _{\mathrm{e}} 3\)
D \(2 \log _3 \mathrm{e}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(2 \log _3 \mathrm{e}\)

Step-by-step Solution

Detailed explanation

\(\begin{aligned}
\mathrm{f}(x) & =\log _{\mathrm{e}} x \\
\mathrm{f}(1) & =\log _{\mathrm{e}} 1=0 \\
\mathrm{f}(3) & =\log _{\mathrm{e}} 3 \text { and } \mathrm{f}^{\prime}(x)=\frac{1}{x}
\end{aligned}\)
By Lagrange's mean value theorem,
\(\begin{aligned}
& f^{\prime}(c)=\frac{f(3)-f(1)}{3-1} \\
& \Rightarrow \frac{1}{c}=\frac{\log _e 3-0}{2} \Rightarrow c=\frac{2}{\log _e 3} \Rightarrow \mathrm{c}=2 \log _3 e
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From MHT CET

Explore more questions on app

	Column I		Column II
i.	\(FADH _2\)	a.	Complex III
ii.	Ubiquinol	b.	Complex IV
iii.	\(NADH + H ^{+}\)	c.	Complex II
iv.	Cytochrome C	d.	Complex I