Relation between Continuity and Differentiability

Q.

The function

$f\left(x\right)=\{\begin{array}{cc}|x-3|,& x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4},& x<1\end{array}$,

is

1. continuous at $x=1$

2. differentiable at $x=1$

3. discontinuous at $x=1$

4. differentiable at $x=3$

Q. Let $S=\left\{\right(\lambda ,\mu )\in R\times R:f(t)=(|\lambda |e^{|t|}-\mu )\cdot \sin (2|t|),t\in R,\text{is a differentiable function}\}.$
Then $S$ is a subset of :

1. $R\times [0,\infty )$
2. $[0,\infty )\times R$
3. $R\times (-\infty ,0)$
4. $(-\infty ,0)\times R$

Q. Let $f\left(x\right)=\{\begin{array}{cc}x{e}^{-(\frac{1}{|x|}+\frac{1}{x})},& x\ne 0\\ 0,& x=0\end{array}$ Then which of the followings is/are correct.

1. $f\left(x\right)$ is continuous at $x=0$
2. $f\left(x\right)$ is differentiable at $x=0$
3. $f\left(x\right)$ is not continuous at $x=0$
4. $f\left(x\right)$ is not differentiable at $x=0$

Q.

$\text{Let f (a) =g (a)= k and their nth derivatives}{\text{f}}^n\left(a\right),{\text{g}}^n\left(a\right)\text{exist and are not equal for some n. Further if}\underset{x\to a}{lim}\frac{f\left(a\right)g\left(x\right)-f\left(a\right)-g\left(a\right)f\left(x\right)+g\left(a\right)}{g\left(x\right)-f\left(x\right)}=4,\text{then the value of k is:}$

1. 4

2. 2

3. 1

4. 0

Q. If x satisfies $|x-1|+|x-2|+|x-3|\ge 6$, then

1. 
2. 
3. 
4. 

Q.

Find the domain and range of the following real functions:

(i) $f\left(x\right)=-|x|$ (ii) $f\left(x\right)=\sqrt{9-x^2}$

Q. Given f(x) is continuos at x₀, for f(x) to be differentiable at x₀, the left hard Derivative and the right hand Derivative must exist fanitely.

1. True
2. False

Q.

The point (2, 3) is a limiting point of a coaxial system of circles of which $x^2+y^2=9$ is a member. The co-ordinates

of the other limiting point is given by

1. 

2. 

3. 

4. 

Q.

Given f(x) is continuos at $x_0$, for f(x) to be differentiable at $x_0$, the left hard Derivative and the right hand Derivative must exist finitely.

1. True

2. False

Q.

Find

$\underset{x\to 1}{\text{lim}}$ f(x) where f(x)= $\{\begin{array}{cc}{x}^2-1,& x\le 1\\ -x^2-1& x>1\end{array}$

Q. If the function$f\left(x\right)$ satisfies $\underset{x\to 1}{lim}\frac{f\left(x\right)-2}{x^2-1}=\pi$ evaluate $\underset{x\to 1}{lim}f\left(x\right)$

Q.

If $f\left(x\right)=|lo{g}_{10}^x|\forall xϵ(0,\infty )$ then

1. f(x) is continuous in $(0,\infty )$
2. f(x) is differentiable in $(0,\infty )$
3. f(x) is continuous but not differentiable in $(0,\infty )$
4. None of these

Q.

The function

$f\left(x\right)=\{\begin{array}{cc}|x-3|,& x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4},& x<1\end{array}$,

is

1. continuous at $x=1$

2. differentiable at $x=1$

3. discontinuous at $x=1$

4. differentiable at $x=3$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\frac{\sqrt{1-\cos 2(x-3)}}{x-3},& x\ne 3\\ k,& x=3\end{array},$ for f(x) to be continuous at x = 3, the value of k

माना $f\left(x\right)=\{\begin{array}{cc}\frac{\sqrt{1-\cos 2(x-3)}}{x-3},& x\ne 3\\ k,& x=3\end{array}$ है, तब x = 3 पर f(x) सतत होने के लिए k का मान

1. Exists and is equal to $\sqrt{2}$
   
   विद्यमान है तथा इसका मान $\sqrt{2}$ है
2. Exists and is equal to $-\sqrt{2}$
   
   विद्यमान है तथा इसका मान $-\sqrt{2}$ है
3. Does not exist because L.H.L is not equal to R.H.L
   
   विद्यमान नहीं है क्योंकि L.H.L का मान R.H.L के बराबर नहीं है
4. Does not exist because L.H.L is not finite
   
   विद्यमान नहीं है क्योंकि L.H.L परिमित नहीं है

Q. The locus of the vertex of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ (a is parameter) is

1. 
2. 
3. 
4. 

Q.

If $f\left(x\right)=|lo{g}_{10}^x|\forall xϵ(0,\infty )$ then

1. f(x) is continuous in

2. f(x) is differentiable in (0, ∞)

3. f(x) is continuous but not differentiable in (0, ∞)

4. None of these

Q. Let $f:R\to R$ be a function such that $|f\left(x\right)|\le x^2$, for all $x\in R$. At $x=0,f$ is

1. Continuous but not differentiable
   
2. Continuous as well as differentiable
3. Neither continuous nor differentiable
4. Differentiable but not continuous

Q.

Find the numerically greatest term in the expansion of $(4+6x{)}^{24}$ when $x=\frac{1}{6}$.

1. 

2. 

3. 

4. 

Q. If $x=1+y+y^2+\cdots \cdots \infty$, then y is

1. 
2. 
3. 
4. 

Q. Let $f\left(x\right)=\frac{\tan \left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}$, where $[.]$ denotes the greatest integer function. Then

1. $f\left(x\right)$ is continuous and differentiable at all $x\in R$
2. $f\left(x\right)$ is continuous at all $x\in R$ but not differentiable at infinitely many points
   
3. $f\left(x\right)$ is neither continuous nor differentiable at a finite number of points
   
4. $f\left(x\right)$ is neither continuous nor differentiable at infinitely many points

Q.

For the expression f(x) = a $x^2$ + bx + c, (a > 0), the condition for both real roots of f(x) to be greater than (or) lesser than a real value M is,

1. f(M) > 0

2. f(M) < 0

3. f(M) ≥ 0

4. f(M) ≤ 0

Q. Let $f\left(x\right)=\{\begin{array}{cc}x{e}^{-(\frac{1}{|x|}+\frac{1}{x})},& x\ne 0\\ 0,& x=0\end{array}$ Then which of the followings is/are correct.

1. $f\left(x\right)$ is continuous at $x=0$
2. $f\left(x\right)$ is differentiable at $x=0$
3. $f\left(x\right)$ is not continuous at $x=0$
4. $f\left(x\right)$ is not differentiable at $x=0$

Q.

Let $f:[-\frac{1}{2},2]\to R$ and $g:[-\frac{1}{2},2]\to R$ be functions defined by
$f\left(x\right)=[x^2-3]\text{and}g\left(x\right)=|x|f\left(x\right)+|4x-7|f\left(x\right)$, where [y] denotes the greatest integer less than or equal to y for $yϵR$. 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. Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $(px+q)(\frac{r}{x}+s)$

Q. Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $\frac{p{x}^2+qx+r}{ax+b}$

Q. Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $\frac{ax+b}{cx+d}$

Q.

Find $\underset{x\to 0}{\text{lim}}$ f(x) where f(x) = $\{\begin{array}{cc}\frac{x}{|x|}& x\ne 0\\ 0& x\ne 0\end{array}$

Q. Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{1}{a{x}^2+bx+c}$

Q. The function f is defined by $f\left(x\right)=\left\{\begin{array}{cc}1-x,& x<0\\ 1,& x=0\\ x+1,& x>0\end{array}\right.$. Draw the graph of f(x).

Q. Statement $1:$ $f\left(x\right)=\sin x+\left[x\right]$ is discontinuous at $x=0,$ where $[.]$ denotes the greatest integer function.
Statement $2:$ If $g\left(x\right)$ is continuous and $h\left(x\right)$ is discontinuous at $x=a,$ then $g\left(x\right)+h\left(x\right)$ will necessarily be discontinuous at $x=a.$