If f(x) = _ - 1 ^x |t| ,dt, x - 1, then - Vidyadip

MCQ

If $f(x) = \int_{ - 1}^x {|t|\,dt,} $ $x \ge - 1,$ then

✓
$f$ and $f'$ are continous for $x + 1 > 0$
B
$f$ is continous but $f'$ is not continous for $x + 1 > 0$
C
$f$ and $f'$ are not continous at $x = 0$
D
$f$ is continous at $x = 0$ but $f'$ is not so

✓

Answer

Correct option: A.

$f$ and $f'$ are continous for $x + 1 > 0$

a
(a) Let us divide the interval into two sub-intervals ${I_1}$, $ - 1 \le x < 0$

so that $x$ is $-ve$ and ${I_2},x \ge 0$ so that $x$ is $+ve.$

For ${I_1},f(x) = \int_{ - 1}^x {( - t)dt = - \frac{1}{2}({x^2} - 1)} $.....$(i)$

For ${I_2},f(x) = \int_{ - 1}^x {( - t)dt + \int_{ 0 }^x {( t)dt }}$

$ = - \frac{1}{2}[{t^2}]_{ - 1}^0 + \frac{1}{2}[{t^2}]_0^x = \frac{1}{2}(1 + {x^2})$.....$(ii)$

Hence the function can be defined as the following

$f(x) = \left\{ \begin{array}{l} - \frac{1}{2}({x^2} - 1),{\rm{If}} - 1 \le x < 0\\\frac{1}{2}({x^2} + 1),{\rm{if}}\,\,x \ge 0\end{array} \right.$ ,

$f'(x) = \left\{ \begin{array}{l} - x,\,\,{\rm{if}} - 1 < x < 0\\\,\,0,\,\,{\rm{if}}\,\,x = 0\\\,\,x,\,{\rm{if}}\,\,x > 0\end{array} \right.$

For $f,\,\,L = R = V = \frac{1}{2}$ at $x = 0$, so $f$ is continuous at $x = 0$.

For $f',\,\,L = R = V = 0$ at $x = 0$, so $f'$ is also continuous at $x = 0$.

Thus both $f$ and $f'$ are continuous at $x = 0$ and hence both are continuous for $x > - 1$

$i.e.,$ $x + 1 > 0$..

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 magnitude of the vector $\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}$ is:

If a line makes angles $\alpha,\beta,\gamma$ with the axis then $\cos 2\alpha+ \cos 2\beta +\cos 2\gamma=$

-2
-1
1
2

The ratio of the rate of flow of water in pipes varies inversely as the square of the radius of the pipes. What is the ratio of the rates of flow in two pipes diameters 2cm and 4cm?

1 : 6
1 : 4
1 : 2
3 : 1

Let $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}=\hat{i}+\hat{j} .$ If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and $\vec{c}$ is $\frac{\pi}{6}$, then the value of $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is:

A plane meets the coordinate axes at A, B, and C such that the centroid of $\triangle{\text{ABC}}$ is the point (a, b, c) if
the eqution of the plane is $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}+\frac{\text{z}}{\text{c}}=\text{k}$,then k =

1
2
3
None of these

The integral $\int_{\, - 1/2}^{\,1/2} {\,\left\{ {[x] + \log \left( {\frac{{1 + x}}{{1 - x}}} \right)} \right\}} \,dx$ equal (where $[.]$ is the greatest integer function)

A non-zero vector $a$ is parallel to the line of intersection of the plane determined by the vectors $i,\,\,i + j$ and the plane determined by the vectors $i - j,\,\,i + k.$ The angle between $a$ and the vector $i - 2j + 2k$ is

A man is known to speak the truth $3$ out of $4$ times. He throws a die and reports that it is a six. The probability that it is actually a six, is

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}=$ __________ + C.

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched

Interval Function