Differentiabilty

Q. If $f\left(x\right)=\{\begin{array}{cc}{2}^{\frac{1}{x}},& forx\ne 0,\\ 3,& forx=0\end{array}$, then

1. f (x) is continuous at x = 0
2. ${lim}_{x\to 0+}f\left(x\right)=0$
3. ${lim}_{x\to 0-}f\left(x\right)=\infty$
4. None of these

Q.

If $f\left(x\right)=|x{|}^3$ , show that $f^{\prime \prime }\left(x\right)$ exists for all real $x$ and find it.

Q. Let $f\left(x\right)=\{\begin{array}{ccc}{x}^2+k,& when& x\ge 0\\ -x^2-k,& when& x<0\end{array}$. If the function f (x) be continuous at x = 0, then k =

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

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{|x-a|}{x-a},& whenx\ne a\\ 1,& whenx=a\end{array}$ , then

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

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{2-(128-6x{)}^{\frac{1}{7}}}{(4x+16{)}^{\frac{1}{4}}-2},& x\ne 0\\ k,& x=0\end{array}$ is continuous at $x=0$ and $g$ is a function with the property that $g(x+y)=g\left(x\right)+g\left(y\right)+xy$ and $\underset{h\to 0}{lim}\frac{g\left(h\right)}{h}=28f\left(0\right)$

1. $g$ is a differentiable function
2. $g\left(x\right)=3x+\frac{x^2}{2}+1$
3. $g\left(x\right)=3x+\frac{x^2}{2}$
4. $g\left(x\right)=3+x$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\frac{x^3+x^2-16x+20}{(x-2{)}^2},& ifx\ne 2\\ & k,ifx=2\end{array}$ f f(x)be continuous for all x, then k =

1. 7
2. -7
3. $\pm$7
4. None of these

Q. If $f\left(x\right)=\{\begin{array}{cc}\text{min}\{x,x^2\},& x\ge 0\\ \text{min}\{2x,x^2-1\},& x<0\end{array}$ then the number of points where $f\left(x\right)$ is non-differentiable in $[-2,2]$ is

Q. If $f\left(x\right)=\{\begin{array}{cc}a{x}^2-b,& when0\le x<1\\ 2,& whenx=1\\ x+1,& when1<x\le 2\end{array}$ is continuous at x = 1, then the most suitable value of a, b are

1. a = 2, b = 0
2. a = 1, b = -1
3. a = 4, b = 2
4. All the above

Q. Each of the following function is defined to be zero at x = 0
$f_1\left(x\right)=x^{2}sgn\left(x\right)$
$f_2\left(x\right)={\int }_0^x{t}^{2}sin\left(\frac{1}{t}\right)dt$
$f_3\left(x\right)=x^{-\frac{1}{3}}|sinx|$
$f_4\left(x\right)=x^3[-x]$
[where sgn(x) denotes signum function and [.] denotes greatest integer function]
$$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{I}& \text{The function}{f}_1& \text{(P)}& \text{continuous but not}\\ & & & \text{differentiable at x = 0}\\ \text{II}& \text{The function}{f}_2\text{is}& \text{(Q)}& \text{first derivative exists at x = 0 but}\\ & & & \text{second derivative does not exist}\\ \text{III}& \text{The function}{f}_3\text{is}& \text{(R)}& \text{second derivative exists at x = 0,}\\ & & & \text{but it is not continuous there}\\ \text{IV}& \text{The function}{f}_4\text{is}& \text{(S)}& \text{second derivative exists at x = 0}\\ & & & \text{and is continuous at x = 0.}\end{array}$$
Which of the following is the only CORRECT combination?

1. (I), (P)
2. (II), (R)
3. (III), (S)
4. (IV), (Q)

Q. If $f\left(x\right)=\{\begin{array}{cc}\text{min}\{x,x^2\},& x\ge 0\\ \text{min}\{2x,x^2-1\},& x<0\end{array}$ then the number of points where $f\left(x\right)$ is non-differentiable in $[-2,2]$ is

Q. If the function $f\left(x\right)=\{\begin{array}{cc}\frac{x^2-(A+2)x+A}{x-2}& forx\ne 2\\ 2& forx=2\end{array}$ is
continuous at $x=2$ , then

1. $A=0$
2. $A=-1$
3. $A=1$
4. $A=2$

Q. The minimum values of $f\left(x\right)=2^{(x^2-3{)}^3+27}$

1. $4$
2. $16$
3. $2^{27}$
4. $1$

Q. Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{cc}max\{-x,x+2\},& x<0\\ 2,& 0\le x<1\\ 3-x,& x\ge 1\end{array}.$ Then

1. $f\left(x\right)$ is continuous for all $x\in R$
2. $f^{\prime }\left(x\right)>0$ for $-1<x<0$
3. $f\left(x\right)$ is non-differentiable at $x=0$
4. $f\left(x\right)$ is discontinuous at $x=0$

Q. The approximate value of$\sqrt{{\left(1.97\right)}^2{\left(4.02\right)}^2{\left(3.98\right)}^2}$

1. $31.59$
2. $5.009$
3. $5.734$
4. $5.099$

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^3,& x<0\\ x^2,& x\ge 0\end{array}$ then

1. $f\left(x\right)$ is continuous at $x=0$
2. $f\left(x\right)$ is differentiable at $x=0$
3. $f^{\prime }\left(x\right)$ is continuous on $R$
4. $f^{\prime \prime }\left(x\right)$ exists for all $xϵR$

Q. If $f\left(x\right)=\text{min}\left\{x^2,-x+1,sgn|-x|\right\}$ then $f\left(x\right)$ is
(where $\text{sgn}\left(x\right)$ denotes signum function of $x$)

1. continuous at $x=0$
2. discontinuous at one point
3. non-differentiable at $3$ points
4. differentiable at $x=\frac{1}{2}$

Q. If $f\left(x\right)=\text{min}\left\{x^2,-x+1,sgn|-x|\right\}$ then $f\left(x\right)$ is
(where $\text{sgn}\left(x\right)$ denotes signum function of $x$)

1. continuous at $x=0$
2. discontinuous at one point
3. non-differentiable at $3$ points
4. differentiable at $x=\frac{1}{2}$

Q. If $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{x-1}{2x^2-7x+5},& x\ne 1\end{array}\\ \begin{array}{cc}-\frac{1}{3},& x=1\end{array}\end{array}$ then $f^{\prime }\left(1\right)$ is equal to

1. $-\left(\frac{2}{9}\right)$
2. $\frac{1}{3}$
3. $-13$
4. $-\left(\frac{1}{9}\right)$

Q. How do you simplify $8^{\frac{4}{3}}$ ?

Q. The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $(0,\frac{\pi }{2})$ is

Q. Consider the function $f\left(x\right)=2\sqrt{6-\left[x\right]}$ at $2<x\le 3$ , where $\left[x\right]$ denotes step up function, then at $x=2$ the function

1. is continuous
2. has missing point removable discontinuity
3. has isolates missing point removable discontinuity
4. has non removable discontinuity finite type

Q. Each of the following function is defined to be zero at x = 0
$f_1\left(x\right)=x^{2}sgn\left(x\right)$
$f_2\left(x\right)={\int }_0^x{t}^{2}sin\left(\frac{1}{t}\right)dt$
$f_3\left(x\right)=x^{-\frac{1}{3}}\left[sinx\right]$
$f_4\left(x\right)=x^3[-x]$
[where sgn(x) denotes signum function and [.] denotes greatest integer function]
$$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{I}& \text{The function}{f}_1& \text{(P)}& \text{continuous but not}\\ & & & \text{differentiable at x = 0}\\ \text{II}& \text{The function}{f}_2\text{is}& \text{(Q)}& \text{first derivative exists at x = 0 but}\\ & & & \text{second derivative does not exist}\\ \text{III}& \text{The function}{f}_3\text{is}& \text{(R)}& \text{second derivative exists at x = 0,}\\ & & & \text{but it is not continuous there}\\ \text{IV}& \text{The function}{f}_4\text{is}& \text{(S)}& \text{second derivative exists at x = 0}\\ & & & \text{and is continuous at x = 0.}\end{array}$$
Which of the following is the only CORRECT combination?

1. (II), (Q)
2. (III), (R)
3. (IV), (S)
4. (I), (P)

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^3,& x<0\\ x^2,& x\ge 0\end{array}$ then

1. $f\left(x\right)$ is continuous at $x=0$
2. $f\left(x\right)$ is differentiable at $x=0$
3. $f^{\prime }\left(x\right)$ is continuous on $R$
4. $f^{\prime \prime }\left(x\right)$ exists for all $xϵR$

Q. Let $f_1\left(x\right)=x{\tan }^{-1}x,f_2\left(x\right)=\left\{\begin{array}{cc}\frac{x}{\ln 2};& x\ne 1\\ 1;& x=1\end{array}\right\},f_3\left(x\right)=\left\{\begin{array}{cc}(x+1)e^{(\frac{1}{|x|}+\frac{1}{x})},& x\ne 0\\ 0,& x=0\end{array}\right\}{f}_4\left(x\right)=\frac{2\tan (\frac{\pi }{4}-x)}{\cot 2x}\text{if}x\ne \frac{\pi }{4}$

$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(I) Number of critical points for}{f}_1\left(x\right)& \text{(P) -1}\\ \text{over its domain is}& \\ \text{(II) Derivative of}{f}_2\left(x\right)\text{at}x=1\text{is}& \text{(Q) 0}\\ \text{(III) Number of points of discountinuity}& \text{(R) 1}\\ \text{for}{f}_3\left(x\right)\text{in the interval [-2, 2]}\text{is}& \\ \text{(IV) Value of}{f}_4\left(\frac{\pi }{4}\right)\text{such that the}& \text{(S) 2}\\ \text{function is continuos everywhere}& \\ \text{in the interval}[\frac{\pi }{6},\frac{\pi }{3}]\text{is}& \\ & \text{(T) 3}\\ & \text{(U) None of these}\end{array}$
Which of the following is only CORRECT combination?

1. $\text{I}\to \text{(S)}$
2. $\text{II}\to \text{(S)}$
3. $\text{III}\to \text{(R)}$
4. $\text{IV}\to \text{(Q)}$

Q. Let $f_1\left(x\right)=x{\tan }^{-1}x,f_2\left(x\right)=\left\{\begin{array}{cc}\frac{x}{\ln 2};& x\ne 1\\ 1;& x=1\end{array}\right\},f_3\left(x\right)=\left\{\begin{array}{cc}(x+1)e^{(\frac{1}{|x|}+\frac{1}{x})},& x\ne 0\\ 0,& x=0\end{array}\right\}{f}_4\left(x\right)=\frac{2\tan (\frac{\pi }{4}-x)}{\cot 2x}\text{if}x\ne \frac{\pi }{4}$

$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(I) Number of critical points for}{f}_1\left(x\right)& \text{(P) -1}\\ \text{over its domain is}& \\ \text{(II) Derivative of}{f}_2\left(x\right)\text{at}x=1\text{is}& \text{(Q) 0}\\ \text{(III) Number of points of discountinuity}& \text{(R) 1}\\ \text{for}{f}_3\left(x\right)\text{in the interval [-2, 2]}\text{is}& \\ \text{(IV) Value of}{f}_4\left(\frac{\pi }{4}\right)\text{such that the}& \text{(S) 2}\\ \text{function is continuos everywhere}& \\ \text{in the interval}[\frac{\pi }{6},\frac{\pi }{3}]\text{is}& \\ & \text{(T) 3}\\ & \text{(U) None of these}\end{array}$
Which of the following is only CORRECT combination?

1. $\text{I}\to \text{(R)}$
2. $\text{II}\to \text{(T)}$
3. $\text{III}\to \text{(U)}$
4. $\text{IV}\to \text{(Q)}$

Q. Prove that ${\int }_{-a}^a{x}^3\sqrt{a^2-x^2}dx=0$