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

Given the function \( f(x) = \log x \), we need to find the number \( C \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \). The expression \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) represents the average rate of change of the function \( f(x) = \log x \) from \( x = e^2 \) to \( x = e^3 \).

First, let's calculate the values of \( f(e^3) \) and \( f(e^2) \):

• \( f(e^3) = \log(e^3) = 3\log e = 3 \) (since \(\log e = 1\))
• \( f(e^2) = \log(e^2) = 2\log e = 2 \)

The average rate of change between \( x = e^2 \) and \( x = e^3 \) is:

\[\frac{f(e^3) - f(e^2)}{e^3 - e^2} = \frac{3 - 2}{e^3 - e^2} = \frac{1}{e^3 - e^2}\]

This implies that the reciprocal of \( C \) is \(\frac{1}{e^3 - e^2}\), which means \( C = e^3 - e^2 \).

Thus, the number \( C \) strictly between \( e^2 \) and \( e^3 \) satisfying the given condition is \( e^3 - e^2 \).