Differentiability in an Interval

Q. Let $f\left(x\right)=\{\begin{array}{cc}-1;& -2\le x<0\\ & \\ x^2-1;& 1\le x\le 2\end{array}$

and $g\left(x\right)=|f\left(x\right)|+f\left(|x|\right)$. Then, in the interval $(-2,2)$, $g$ is :

1. not continuous
2. not differentiable at one point
3. not differentiable at two points
4. differentiable at all points

Q.

If $f\left(x\right)=x^2-1$ and $g\left(x\right)={\left(x+1\right)}^2$, then $\left(gof\right)\left(x\right)$ is equal to

1. ${\left(x+4\right)}^4-1$

2. $x^4-1$

3. $x^4$

4. ${\left(x+1\right)}^4$

5. ${\left(x-1\right)}^4-1$

Q.

$x^x$ has a stationary point at

1. $x=e$

2. $x=\frac{1}{e}$

3. $x=1$

4. $x=\sqrt{e}$

Q. A function $f$ is defined on $[-3,3]$ as
$f\left(x\right)=\{\begin{array}{c}\text{min}\{|x|,2-x^2\},-2\le x\le 2\\ \left[|x|\right],2<|x|\le 3\end{array}$
where $\left[x\right]$ denotes the greatest integer $\le x$. The number of points, where $f$ is not differentiable in $(-3,3)$ is

Q. $f:R\to R$ given by $f\left(x\right)=x^3+3x^2+12x-2\sin x,$ then $f\left(x\right)$ is

1. One-one function
2. Many-one function
3. Constant function
4. None of these

Q. If f is a real- valued differentiable function satisfying $|f\left(x\right)-f\left(y\right)|\le (x-y{)}^2,x,yϵRandf(0)=0,thenf(1)$ equal

1. 1
2. -1
3. 2
4. 0

Q. Let $a,b\in R$ and $f:R\to R$ be defined by $f\left(x\right)=a\cos (|x^3-x|)+b|x|\sin (|x^3+x|).$ Then $f$ is

1. differentiable at $x=0$ if $a=0$ and $b=1$
2. differentiable at $x=1$ if $a=1$ and $b=0$
3. NOT differentiable at $x=0$ if $a=1$ and $b=0$
4. NOT differentiable at $x=1$ if $a=1$ and $b=1$

Q. The minimum possible value of
$|z{|}^2+|z-3{|}^2+|z-6i{|}^2(z\in C)$ is

1. $45$
2. $15$
3. $60$
4. $30$

Q. Let $f$ be a differentiable function satisfying $f(x+2y)=2yf\left(x\right)+xf\left(y\right)-3xy+1\forall x,yϵR$ such that $f^{\prime }\left(0\right)=1$ then $f\left(2\right)$ is

1. 1
2. 4
3. 5
4. 3

Q. The number of points at which the function f(x) = |x - 0.5| + |x - 1| + tan x does not have a derivative in the interval (0, 2), is
[MNR 1995]

1. 3
2. 4
3. 1
4. 2

Q. Let $f(x+y)=f\left(x\right)+f\left(y\right)+2xy-1\forall x,y\in R$. If $f\left(x\right)$ is differentiable and $f^{\prime }\left(0\right)=\sin \varphi$, then

1. $f\left(x\right)>0\forall x\in R$
2. $f\left(x\right)<0\forall x\in R$
3. $f\left(x\right)$ is linear
4. $f\left(x\right)=\sin \varphi \forall x\in R$

Q. Find the intervals in which the function $f$ given by $f\left(x\right)=2x^2-3x$ is

$\left(a\right)$ increasing
$\left(b\right)$ decreasing

Q. $\text{Let}f:[-\frac{1}{2},2]\to R\text{and}g:[-\frac{1}{2},2]\to R$ be functions defined by $f\left(x\right)=[x^2-3]$ and $g\left(x\right)=|x|f\left(x\right)+|4x-7|f\left(x\right),$ where $\left[y\right]$ denotes the greatest integer less than or equal to $y$ for $y\in R.$ Then

1. $f$ is discontinuous exactly at three points in $[-\frac{1}{2},2]$
2. $f$ is discontinuous exactly at four points in $[-\frac{1}{2},2]$
3. $g$ is NOT differentiable exactly at four points in $(-\frac{1}{2},2)$
4. $g$ is NOT differentiable exactly at five points in $(-\frac{1}{2},2)$

Q. From given functions, which of the following(s) is a point function

1. $\sqrt{2x-4}+\sqrt{4-2x}$
2. $\sqrt{3x-6}+\sqrt{3-x}$
3. $\sqrt{2x-4}+\ln (2-x)$
4. $\sqrt{\text{sgn}(-x^2)}$

Q. The function $f\left(x\right)={\cos }^{-1}(4x^3-3x)$ is

1. always differentiable
2. not continuous at 2 points only
3. not differentiable at 2 points only
4. not differentiable at 4 points only

Q. Let f(x) be a continuous and differentiable function and f(y)f(x + y) = f(x) for all real values of x and y. If f(5) = 3 and f'(3) = 7 then the value of f'(8) is

1. $\frac{7}{3}$
2. $7$
3. $0$
4. $\frac{1}{7}$

Q. The function $f\left(x\right)=x^{\frac{1}{3}}(x-1)$

1. Has one stationary point
2. Has two critical points
3. Has two stationary points
4. Has one point of local minima

Q. If $f\left(x\right)=x|x^2-3x|e^{|x-3|}+[\frac{3+sgn(x^2-1-5x)}{4+x^2}]+\{\frac{1}{3}+[\frac{x^2-2x}{|x|+1}]\}$ is non differentiable at $x=x_1,x_2,x_3.....x_n$ then ${\sum }_{i=1}^n{x}_i$ equals
( where $[.],\{.\},sgn(.)$, denotes greatest integer function, fractional part function and signum function respectively)

1. 0
2. 3
3. 4
4. 5

Q.

The set of all those points, where the function

$f\left(x\right)=\frac{x}{1+|x|}$

is differentiable, is

1. $(-\infty ,\infty )$

2. [0, $\infty$]

3. $(-\infty ,0)\cup (0,\infty )$

4. $(0,\infty )$

Q. For the matrices $A$ and $B$, verify that ${\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }$ where
(i) $A=\left[\begin{array}{c}1\\ -4\\ 3\end{array}\right],B=\left[\begin{array}{ccc}-1& 2& 1\end{array}\right]$
(ii) $A=\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right],B=\left[\begin{array}{ccc}1& 5& 7\end{array}\right]$

Q. The number of points at which $f\left(x\right)=\{\begin{array}{cc}min(x,x^2),& \text{if}-\infty <x<1\\ min(2x-1,x^2),& \text{if}x\ge 1\end{array}$ is not differentiable is

1. $1$
2. $3$
3. $2$
4. $0$

Q. Represent the given complex numbers in polar form $\frac{1}{1+i}$.

Q.

The set of all those points, where the function

$f\left(x\right)=\frac{x}{1+|x|}$

is differentiable, is

1. [0, $\infty$]

2. $(-\infty ,\infty )$

3. $(-\infty ,0)\cup (0,\infty )$

4. $(0,\infty )$

Q. Let $f\left(x\right)=x^2-6x+5$ and $m$ is the number of points of non-derivability of $y=|f\left(|x|\right)|.$ If $|f\left(|x|\right)|=k,k\in R$ has at least $m$ distinct solution(s), then the number of integral values of $k$ is

1. $2$
2. $3$
3. $4$
4. $5$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\left\{x^2\right\},& -1\le x<1\\ |1-2x|,& 1\le x<2\\ (1-x^2)\text{sgn}(x^2-3x-4),& 2\le x\le 4\end{array}$

where $\left\{k\right\}$ and $\text{sgn}\left(k\right)$ denote fractional part function and signum function of $k$ respectively. If $m$ denotes the number of points of discontinuity of $f\left(x\right)$ in $[-1,4]$ and $n$ denotes the number of points of non-differentiability of $f\left(x\right)$ in $(-1,4),$ then $(m+n)$ is equal to

1. $2$
2. $4$
3. $5$
4. $3$

Q. Let $\overrightarrow{a}=-\hat{i}-\hat{k},\overrightarrow{b}=-\hat{i}+\hat{j}$ and $\overrightarrow{c}=\hat{i}+2\hat{j}+3\hat{k}$ be three given vectors. lf $\overrightarrow{r}$ is a vector such that $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}$ and $\overrightarrow{r}.\overrightarrow{a}=0$, then the value of $\overrightarrow{r}.\overrightarrow{b}$ is

1. $3$
2. $6$
3. $9$
4. $12$

Q. If $f\left(x\right)=\{\begin{array}{ccc}\left(\frac{x^2}{a}\right)-a,& when& x<a\\ 0,& when& x=a\\ a-\left(\frac{x^2}{a}\right),& when& x>a\end{array}$, then

1. f(x) is continuous at x = a
2. $li{m}_{x\to a}f\left(x\right)=a$
3. f(x) is discontinuous at x = a
4. None of these

Q. The set of all points for which $f\left(x\right)=\frac{|x-3|}{|x-2|}+\frac{1}{1+\left[x\right]}$ is continuous, is $($where $[\ast ]$ represents the greatest integer function$)$

1. $R-[-1,0]$
2. $R$
3. $R-\left\{\right(-1,0)\cup n,n\in I\}$
4. $R-\left(\right\{2\}\cup [-1,0])$

Q. Differentiate $x^2-3x+6$ w.r.t $\frac{1}{x}$

Q. Let $f:R\to (0,\infty )$ satisfy the relation $f(x+y)=f\left(x\right)f\left(y\right),f\left(x\right)\ne 0$ for any $x$ and $f\left(x\right)$ is differentiable on $R$ such that $f^{\prime }\left(0\right)=\ln 2.$

	List-I		List-II
(A)	If $\left|\begin{array}{ccc}f\left(x\right)& 1& 0\\ 2& f\left(x\right)& 1\\ 3& 1& 2\end{array}\right|=0,$ then $x$ equals	(P)	$0$
(B)	$\underset{x\to 0}{\text{lim}}\frac{f\left(x\right)-1}{x}$ equals	(Q)	$1$
(C)	Number of points of non-differentiability of $g\left(x\right)=\text{min}\left\{f\right(x),5x-2\}$ is	(R)	$2$
(D)	Number of points of non-differentiability of $h\left(x\right)=\text{min}\left\{f\right(x),[x\left]\right\},$ $x\in (-1,3),$ where $[.]$ denotes the greatest integer function is	(S)	$3$
		(T)	$\ln 2$

Which of the following is CORRECT combination ?