If c is a point at which Rolle's theorem holds for the function, … - Vidyadip

MCQ

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

A
$\frac{\sqrt{3}}{7}$
✓
$\frac{1}{12}$
C
$-\frac{1}{24}$
D
$-\frac{1}{12}$

✓

Answer

Correct option: B.

$\frac{1}{12}$

b
$\frac{9+\alpha}{21}=\frac{16+\alpha}{28} \Rightarrow \alpha=12$

Also, $f^{\prime}(\mathrm{x})=\frac{7 \mathrm{x}}{\mathrm{x}^{2}+12} \times \frac{\mathrm{x}^{2}-12}{7 \mathrm{x}^{2}}=\frac{\mathrm{x}^{2}-12}{\mathrm{x}\left(\mathrm{x}^{2}+12\right)}$

Hence, $c=2 \sqrt{3}$

Now, $f^{\prime \prime}(c)=\frac{1}{12}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate: $\int \frac{\left(x^4-x\right)^{\frac{1}{4}}}{x^5} d x$

The solution of the differention equation $(1+\text{x}^{2})\frac{\text{dy}}{\text{dx}}+1+\text{y}^{2}=0$ is:

$\tan^{-1}\text{x}-\tan^{-1}\text{y}=\tan^{-1}\text{C}$
$\tan^{-1}\text{y}-\tan^{-1}\text{x}=\tan^{-1}\text{C}$
$\tan^{-1}\text{y}\pm\tan^{-1}\text{x}=\tan^{-1}\text{C}$
$\tan^{-1}\text{y}+\tan^{-1}\text{x}=\tan^{-1}\text{C}$

lf $\text{AB}\perp\text{BC}$ then the value of $\lambda$ equal, where A(2k, 2, 3), B(k, 1, 5), C(3 + k, 2, 1):

$3$
$\frac{1}{3}$
$-3$
$-\frac{1}{3}$

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$

Objective of LPP is:

Let $f, g: R \rightarrow R$ be two real valued functions defined as $f(x)=\left\{\begin{array}{cl}-|x+3| & , \quad x<0 \\ e^{x} & , \quad x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x & , \quad x<0 \\ 4 x+k_{2} & , \quad x \geq 0\end{array}\right.$, where $k_{1}$ and $k_{2}$ are real constants. If $(gof)$ is differentiable at $x=0$, then $(gof) (-4)+(gof)\, (4)$ is equal to

$\int_{}^{} {\frac{{dx}}{{\tan x + \cot x}}} = $

Consider a $\triangle \mathrm{ABC}$ where $\mathrm{A}(1,3,2), \mathrm{B}(-2,8,0)$ and $\mathrm{C}(3,6,7)$. If the angle bisector of $\angle \mathrm{BAC}$ meets the line $B C$ at $D$, then the length of the projection of the vector $\overrightarrow{A D}$ on the vector $\overrightarrow{A C}$ is:

$\int_{}^{} {{{(\tan x - \cot x)}^2}\;dx = } $

If a dice is thrown $7$ times, then the probability of obtaining $5$ exactly $4$ times is