Question

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

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

Hint:

In L.M.V.T. problems, always verify the value of $c$ lies within the given interval.

Answer 1

The Correct Option is B

Step 1: Applying Lagrange’s Mean Value Theorem.
According to L.M.V.T., there exists a point $c \in (1,3)$ such that \[ f'(c) = \frac{f(3) - f(1)}{3 - 1} \]
Step 2: Computing $f'(x)$.
\[ f(x) = x + \frac{1}{x} \Rightarrow f'(x) = 1 - \frac{1}{x^2} \]
Step 3: Evaluating the right-hand side.
\[ f(3) = 3 + \frac{1}{3} = \frac{10}{3}, \quad f(1) = 1 + 1 = 2 \] \[ \frac{f(3) - f(1)}{2} = \frac{\frac{10}{3} - 2}{2} = \frac{2}{3} \]
Step 4: Solving for $c$.
\[ 1 - \frac{1}{c^2} = \frac{2}{3} \Rightarrow \frac{1}{c^2} = \frac{1}{3} \Rightarrow c^2 = 3 \] Since $c \in (1,3)$, \[ c = \sqrt{3} \]
Step 5: Conclusion.
The value of $c$ is $\sqrt{3}$.