Differentiability in an Interval

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. \N
2. 3
3. 4
4. 5

Q. If $(x+2),3,5$ are the lengths of sides of a triangle, then $x$ lies in

1. $(0,6)$
2. $(-4,6)$
3. $(-1,6)$
4. $(1,6)$

Q. If f(x) is a differentiable function such that F : R $\to$ R and $f\left(\frac{1}{n}\right)=0\forall n\le 1,nϵI$ then
[IIT Screening 2005]

1. $f\left(x\right)=0\forall xϵ(0,1)$
2. f(0) = 0 = f'(0)
3. f(0) = 0 but f (0) may or may not be 0
4. $|f\left(x\right)|\le 1\forall xϵ(0,1)$

Q. Let $f\left(x\right)=\frac{x}{1+x^2}$ and $g\left(x\right)=\frac{e^{-x}}{1+\left[x\right]}$, where $[.]$ represents the greatest integer function. Then the number of integral value(s) of $x$ which are not lying in the domain of $f+g$ is

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

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. The number of integral elements in the range of the function $f\left(x\right)=\frac{e^x-\log x}{e^x+\log x};x>1$ is

Q. If $f\left(x\right)=\{\begin{array}{cc}\left[x\right],& -2\le x\le -\frac{1}{2}\\ 2x^2-1,& -\frac{1}{2}<x\le 2\end{array};$ where $[.]$ represents the greatest integer function, then the number of point(s) of discontinuity of $f\left(x\right)$ is

1. $1$
2. continuous at every point in domain
3. $2$
4. $3$

Q. If $f\left(x\right)=\{\begin{array}{cc}a{e}^{-ax},& x\le 0\\ x^3-4x+2b,& x>0\end{array}$

is differentiable at $x=0$, then the value of $|a|+\sqrt{b^2-1}$ is

1. $\sqrt{3}$
2. $2$
3. $0$
4. None of the above

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 $[.]$ denotes the greatest integer function. The number of points, where $f$ is not differentiable in $(-3,3)$ is

Q. The range of $a\in R$ for which the function
$f\left(x\right)=(4a-3)(x+{\log }_{e}5)+2(a-7)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right),$ $x\ne 2n\pi ,n\in N$ has critical points, is:

1. $[-\frac{4}{3},2]$
2. $[1,\infty )$
3. $(-3,1)$
4. $(-\infty ,-1]$

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. 4
2. 1
3. 5
4. 3

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

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

Q. Consider the function $f\left(x\right)=x^2+mx+n,$ where $D=m^2-4n>0.$
$\begin{array}{cc}Column1& Column2\\ \text{Condition on}m\text{and}n& \text{Number of points of non-}\\ & \text{differentiability of}g\left(x\right)=|f\left(|x|\right)|\\ \text{a.}m<0,n>0& \text{p.}1\\ \text{b.}n=0,m<0& \text{q.}2\\ \text{c.}n=0,m>0& \text{r.}3\\ \text{d.}m=0,n<0& \text{s.}5\end{array}$
Then which of the following is correct ?

1. $a\to s,b\to r,c\to p,d\to q$
2. $a\to r,b\to s,c\to p,d\to q$
3. $a\to s,b\to r,c\to q,d\to p$
4. $a\to s,b\to p,c\to r,d\to q$

Q. If $f\left(x\right)=\{\begin{array}{cc}x,& x\le 1\\ x^2+bx+c,& x>1\end{array}$
is a differentiable function, then the value of $5c-8b$ is

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. 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. 2
2. 1
3. -1
4. \N

Q. If $f\left(x\right)=\{\begin{array}{cc}x,& x\le 1\\ x^2+bx+c,& x>1\end{array}$
is a differentiable function, then the value of $5c-8b$ is

Q. If $f\left(x\right)=|1-x|$, then the points where $si{n}^{-1}\left(f|x|\right)$ is non-differentiable are

1. $\{0,1\}$
2. $\{0,-1\}$
3. $\{0,1,-1\}$
4. None of these