The function f(x) = [ * 20 c | x , - ,3 | , , , , , , , , , , , ,… - Vidyadip

MCQ

The function $f(x) = \left[ {\begin{array}{*{20}{c}} {\left| {x\, - \,3} \right|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}&{,\,\,\,\,x\, \geqslant \,1} \\ {\left( {\tfrac{{{x^2}}}{4}} \right)\, - \,\left( {\tfrac{{3\,x}}{2}} \right)\, + \,\left( {\tfrac{{13}}{4}} \right)}&{,\,\,\,\,x\, < \,1} \end{array}} \right.$ is :

A
continuous at $x = 1$
B
diff. at $x = 1$
C
continuous at $x = 3$
✓
All of the above

✓

Answer

Correct option: D.

All of the above

d
$f (x) = \left[ \begin{gathered} x - 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,x \geqslant \,3 \hfill \\ 3 - x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,1 \leqslant x < 3 \hfill \\ \frac{{{x^2}}}{4}\, - \,\frac{{3x}}{2}\, + \frac{{13}}{4}\,\,if\,x\, < 1 \hfill \\ \end{gathered} \right.$
$f ‘(1^+) =$ $\mathop {Limit}\limits_{h \to 0} \frac{{f(1 + h) - f(1)}}{h}\,\,$

= $\mathop {Limit}\limits_{h \to 0} \frac{{3 - (1 + h) - 2}}{h}\,\, $

$= \, - 1$
$f ‘(1^-) =$ $\mathop {Limit}\limits_{h \to 0} \frac{{\frac{{{{(1 - h)}^2}}}{4}\,\, - \,\,\frac{3}{2}\,(1 - h)\, + \frac{{13}}{4}\, - 2}}{{ - \,\,h}}$

=$\mathop {Limit}\limits_{h \to 0} \frac{{{{(1 - h)}^2}\, - 6(1 - h)\, + 5}}{{ - 4h}}\,$
=$\mathop {Limit}\limits_{h \to 0} \frac{{{h^2} - 2h\, + 6h}}{{ - 4h}}\,\,$

$= - 1$
$\Rightarrow\,\, f$ is continuous at $x =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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let ${I_1} = \int_a^{\pi - a} {xf(\sin x)dx,\,{I_2} = \int_a^{\pi - a} {\,\,f(\sin x)dx} } $, then ${I_2}$ is equal to

${{{d^2}} \over {d{x^2}}}(2\cos x\,\cos 3x) = $

Let $x_{1}, x_{2}, \ldots \ldots x_{10}$ be ten observations such that $\sum_{i=1}^{10}\left(x_{i}-2\right)=30, \sum_{i=1}^{10}\left(x_{i}-\beta\right)^{2}=98, \beta>2$ and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^{2}$ are respectively the mean and the variance of $2\left(x_{1}-1\right)+4 \beta, 2\left(x_{2}-1\right)+$ $4 \beta, \ldots ., 2\left(x_{10}-1\right)+4 \beta$, then $\frac{\beta \mu}{\sigma^{2}}$ is equal to :

→

The general solution of $sin\, x + sin \,5x = sin\, 2x + sin \,4x$ is :

There are $15$ players in a cricket team, out of which $6$ are bowlers, $7$ are batsmen and $2$ are wicketkeepers. The number of ways, a team of $11$ players be selected from them so as to include at least $4$ bowlers, $5$ batsmen and $1$ wicketkeeper, is $.....$

For the circle ${x^2} + {y^2} + 6x - 8y + 9 = 0$, which of the following statements is true

An ellipse having foci at $(3, 1)$ and $(1, 1) $ passes through the point $(1, 3),$ then its eccentricity is

If $A = \left[ {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right]$, then ${A^4}$ is equal to

The line $4x -3y + 15 = 0$ intersect the circle $x^2 + y^2 -6x -8y = 0$ at two points $A$ and $B$. Maximum area of $\Delta ABC$, where $C$ is a point on circumference of circle, will be - .............. $\mathrm{sq.\, units}$

If $\alpha ,\beta \ne 0$ and $f\left( n \right) = {\alpha ^n} + {\beta ^n}$ and $\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 - \alpha } \right)^2}$ ${\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2}$ ,then $K=$ . . . . . .