Question

If the L.M.V.T. holds for the function \( f(x) = x + \dfrac{1}{x} \), \(x \in [1,3]\), then \(c =\)

• A. \( \sqrt{3} \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( -\sqrt{3} \)

Hint:

In L.M.V.T. problems, always equate \(f'(c)\) to the average rate of change over the interval.

Answer 1

The Correct Option is A

Step 1: Apply Lagrange’s Mean Value Theorem.
According to L.M.V.T., there exists \(c \in (1,3)\) such that \[ f'(c) = \frac{f(3) - f(1)}{3 - 1} \]
Step 2: Compute \(f(3)\) and \(f(1)\).
\[ f(3) = 3 + \frac{1}{3} = \frac{10}{3}, \quad f(1) = 1 + 1 = 2 \]
Step 3: Evaluate the right-hand side.
\[ \frac{\frac{10}{3} - 2}{2} = \frac{4/3}{2} = \frac{2}{3} \]
Step 4: Find the derivative and solve.
\[ f'(x) = 1 - \frac{1}{x^2} \] \[ 1 - \frac{1}{c^2} = \frac{2}{3} \Rightarrow \frac{1}{c^2} = \frac{1}{3} \Rightarrow c = \sqrt{3} \]