A continuous and differentiable function ‘ f ‘ satisfies the cond… - Vidyadip

MCQ

A continuous and differentiable function $‘ f ‘$ satisfies the condition ,$\int\limits_0^x \, f (t) d t = f^2 (x) - 1$ for all real $‘ x ‘$. Then :

A
$‘ f ‘$ is monotonic increasing $\forall x \in R$
B
$‘ f ‘$ is monotonic decreasing $\forall x \in R$
C
the graph of $y = f (x)$ is a straight line.
✓
both $(A)$ and $(C)$

✓

Answer

Correct option: D.

both $(A)$ and $(C)$

d
Differentiating $f (x) = 2 f (x). f ‘ (x) $

$\Rightarrow f ‘ (x) =$ $\frac{1}{2}$ $( f (x)) \ne 0)$
Hence $f (x) =$ $\frac{x}{2}$ $+ c$.

Put $x = 0$ ; $f (0) = c $; but $f^2 (0) = 1$
$\Rightarrow f (0) = ± 1$
Hence $f (x) = \frac{x}{2} ± 1$

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 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The shortest distance between the lines $\frac{\text{x}-3}{3}=\frac{\text{y}-8}{-1}=\frac{\text{z}-3}{1}$ and, $\frac{\text{x}+3}{-3}=\frac{\text{y}+7}{2}=\frac{\text{z}-6}{4}$ is:

Value of $\left| {\begin{array}{*{20}{c}}
{{{(b + c)}^2}}&{{a^2}}&{{a^2}} \\
{{b^2}}&{{{(a + c)}^2}}&{{b^2}} \\
{{c^2}}&{{c^2}}&{{{(a + b)}^2}}
\end{array}} \right|$ is equal to

If $\vec{a}, \vec{b}$ and $(\vec{a}+\vec{b})$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is :

Evaluate: $\left|\begin{array}{ll}\cos 15^{\circ} & \sin 15^{\circ} \\ \sin 75^{\circ} & \cos 75^{\circ}\end{array}\right|$

If $x = \sin t\cos 2t$ and $y = \cos t\sin 2t$, then at $t = {\pi \over 4},$ the value of ${{dy} \over {dx}}$ is equal to

The system of equations $(\sin\theta ) x + 2z = 0$ , $(\cos\theta ) x + (\sin\theta )y = 0$ , $(\cos\theta )y + 2z = a$ has

If $\vec{\text{a}}$ and $\vec{\text{b}}$ be two unit vectors and $\theta$ the angle between them, than $\vec{\text{a}}+\vec{\text{b}}$ is a unit vector if $\theta=$

$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$
$\frac{2\pi}{3}$

If the fucnction $\text{f(x)}=\begin{cases}(\cos\text{x})^{\frac{1}{\text{x}}},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$ is continuouse at x = 0, then the value of k is:

0
1
-1
e

Area between the parabolas y² = 4ax and x² = 4ay is:

$\frac{2}{3}\text{a}^2-5$
$\frac{15}{4}\text{a}^2+5$
$\frac{16}{3}\text{a}^2+2$
$\frac{16}{3}\text{a}^2$

If $\int_{}^{} {\frac{{2x + 3}}{{(x - 1)({x^2} + 1)}}\;dx = {{\log }_e}\left\{ {{{(x - 1)}^{\frac{5}{2}}}{{({x^2} + 1)}^a}} \right\}} - \frac{1}{2}{\tan ^{ - 1}}x + A$,where $A$ is any arbitrary constant, then the value of $a$ is