Question

Let \( f(x) = \log x \). The number \( C \) strictly between \( e^2 \) and \( e^3 \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) is

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

Hint:

Apply MVT and relate derivatives to expressions involving function values.

Answer 1

The Correct Option is D

Apply the Mean Value Theorem (MVT) for differentiable function \( f(x) = \log x \) on interval \([e^2, e^3]\). There exists \( C \in (e^2, e^3) \) such that: \[ f'(C) = \frac{f(e^3) - f(e^2)}{e^3 - e^2} = \frac{3\log e - 2\log e}{e^3 - e^2} = \frac{1}{e^3 - e^2} \] Since \( f'(x) = \frac{1}{x} \), this implies \( \frac{1}{C} = \frac{1}{e^3 - e^2} \Rightarrow C = e^3 - e^2 \)