If c is a point at which Rolle's theorem holds for the function

Question:

If $c$ is a point at which Rolle's theorem holds for the function, $f(x)=\log _{e}\left(\frac{x^{2}+a}{7 x}\right)$ in the interval $[3,4]$, where $\alpha \in R$, then $f^{\prime \prime}(c)$ is equal to:

  1. (1) $-\frac{1}{12}$

  2. (2) $\frac{1}{12}$

  3. (3) $-\frac{1}{24}$

  4. (4) $\frac{\sqrt{3}}{7}$


Correct Option: , 2

Solution:

Since, Rolle's theorem is applicable

$\therefore f(a)=f(b)$

$f(3)=f(4) \Rightarrow \alpha=12$

$f^{\prime}(x)=\frac{x^{2}-12}{x\left(x^{2}+12\right)}$

As $f^{\prime}(c)=0$ (by Rolle's theorem)

$x=\pm \sqrt{12}, \quad \therefore \quad c=\sqrt{12}, \therefore \quad f^{\prime \prime}(c)=\frac{1}{12}$

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