JEE PYQ: Application Of Derivatives Question 47
Question 47 - 2020 (08 Jan Shift 1)
If $c$ is a point at which Rolle’s theorem holds for the function $f(x) = \log_e\left(\frac{x^2 + \alpha}{7x}\right)$ in the interval $[3, 4]$, where $\alpha \in \mathbb{R}$, then $f’’(c)$ is equal to:
(1) $-\frac{1}{12}$
(2) $\frac{1}{12}$
(3) $-\frac{1}{24}$
(4) $\frac{\sqrt{3}}{7}$
Show Answer
Answer: (2)
Solution
$f(3) = f(4) \Rightarrow \frac{9+\alpha}{21} = \frac{16+\alpha}{28} \Rightarrow \alpha = 12$. $f’(c) = 0 \Rightarrow \frac{x^2 - 12}{x(x^2+12)} = 0 \Rightarrow c = \sqrt{12}$. $f’’(c) = \frac{1}{12}$.
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