Question

If Lagrange's mean value theorem is applied to the function $$ f(x) = e^x $$ defined on the interval $ [1, 2] $ and the value of $ c \in (1, 2) $ is $ k $, then find $ e^{k-1} $.

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

Hint:

Apply mean value theorem formula carefully.

Answer 1

The Correct Option is B

By Lagrange's Mean Value Theorem, there exists \( k \in (1,2) \) such that: \[ f'(k) = \frac{f(2) - f(1)}{2 - 1} \] Given \( f(x) = e^x \), so \( f'(x) = e^x \): \[ e^k = e^2 - e^1 = e^2 - e \] Divide both sides by \( e \): \[ e^{k-1} = e - 1 \]