Relation between Continuity and Differentiability

Q.

What if the determinant is zero?

Q. Find the derivative of the function $f\left(x\right)=2x^2+3x-5$ at $x=-1$. Also prove that $f^{\prime }\left(0\right)+3f^{\prime }(-1)=0$

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. Let $f=\left\{\right(1,1),(2,3),(0,-1),(-1,-3\left)\right\}$ be a linear function from $Z$ into $Z.$ Find $f\left(x\right)$

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. 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. Let $\frac{dy}{dx}-2y\cot x=\cos x$ such that $y\left(\frac{\pi }{2}\right)=0$. If the maximum value of $y$ is $k,$ then the value of $k$ is

Q.

If function be $f\left(x\right)=\{\begin{array}{cc}\frac{x^2-1}{x-1}& \text{when}x\ne 1\\ k& \text{when}x=1\end{array}$ is continuous at $x=1$, then the value of $k$ is?

1. $-1$
2. $-2$
3. $2$
4. $-3$

Q. If $f\left(x\right)=\{\begin{array}{cc}{x}^2\left(\text{sgn}\right[x\left]\right)+\left\{x\right\},& 0\le x<2\\ \sin x+|x-3|,& 2\le x<4\end{array},$
where [.] and {.} represent the greatest integer and the fractional part function, respectively. Then

1. $f\left(x\right)$ is differentiable at $x=1$
2. $f\left(x\right)$ is continuous but non-differentiable at $x=1$
3. $f\left(x\right)$ is non-differentiable at $x=2$
4. $f\left(x\right)$ is discontinuous at $x=2$

Q. The number of point(s) of non-differentiability for $f\left(x\right)=\left[e^x\right]+|x^2-3x+2|\text{in}(-1,3)$ is ( where [.] denotes greatest integer function, $e^3=20.1$ )

Q. If ${\sin }^{-1}x+|y|=2y$, then $y$ as a function of $x$ is

1. defined for $-1\le x\le 1$
2. continuous at $x=0$
3. differentiable for all $x$ in $(-1,1)$
4. such that $\frac{dy}{dx}=\frac{1}{3\sqrt{1-x^2}}$ for $-1<x<0$

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. 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. False
2. True

Q.

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

(i)

(ii)

(iii)

Q. Let $f\left(x\right)=\{\begin{array}{cc}\frac{1}{|x|}:& |x|\ge 1\\ a{x}^2+b:& |x|<1\end{array}$ be continuous and differentiable every where. Then a and b are:

1. $\frac{-1}{2},\frac{3}{2}$
2. $\frac{1}{2},\frac{-3}{2}$
3. $\frac{1}{2},\frac{3}{2}$
4. $\frac{1}{3},\frac{1}{2}$

Q. At the point x = 1, the given function $f\left(x\right)=\{\begin{array}{cc}{x}^3-1;& 1<x<\infty \\ x-1;& -\infty <x\le 1\end{array}$ is

1. Discontinuous and not differentiable
2. Continuous and differentiable
3. Continuous and not differentiable
4. Discontinuous and differentiable

Q. If $f\left(x\right)=(x-1{)}^2,$ then which of the following is/are true

1. $f$ is continuous at $x=1$
2. $f$ is not derivable at $x=1$
3. $f$ is discontinuous at $x=1$
4. $f$ is derivable at $x=1$

Q.

Find the value of $\underset{x\to \frac{\pi }{4}}{lim}\frac{cosx-sinx}{cos2x}$

1. limit does not exist

2. 

3. 

4. 

Q. If $f\left(x\right)=\{\begin{array}{c}1,when0<x\le \frac{3\pi }{4}\\ 2sin\frac{2}{9}x,when\frac{3\pi }{4}<x<\pi \end{array}$ , then

1. f(x) is continuous at $x=\frac{3\pi }{4}$
2. f(x) is discontinuous at $x=\frac{3\pi }{4}$
3. f(x) is continuous at x = 0
4. f(x) is continuous at x = $\pi$

Q. Let $f\left(x\right)=x|x|,g\left(x\right)=\sin \left(x\right)$ and $h\left(x\right)=(g\circ f)\left(x\right)$. Then

1. $h\left(x\right)$ is not differentiable at $x=0.$
   
2. $h^{\prime }\left(x\right)$ is continuous but not differentiable at $x=0.$
   
3. $h\left(x\right)$ is differentiable at $x=0,$ but $h^{\prime }\left(x\right)$ is not continuous at $x=0.$
   
4. $h^{\prime }\left(x\right)$ is differentiable at $x=0.$

Q. If $f\left(x\right)=\{\begin{array}{cc}x-1& x<0\\ \frac{1}{4},& x=0\\ x^2,& x>0\end{array}$ , then

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

Q. Consider $f\left(x\right)=sgn\left(sinx\right)+\left\{x\right\};2\le x\le 4$ and $g\left(x\right)=-2+\mid x-3\mid ;$ where $\{.\}$ denotes fractional part function and $sgn$ denotes signum function. Then $\underset{x\to 3}{lim}gof\left(x\right)$ is equal to

1. -1
2. 0
3. 1
4. does not exist

Q. The function defined by $f\left(x\right)=\{\begin{array}{cc}|x-3|;& x\ge 1\\ \frac{1}{4}{x}^2-\frac{3}{2}x+\frac{13}{4};& x<1\end{array}$ is

1. Continuous at x = 3
2. Continuous at x = 1
3. Differentiable at x = 1

4. All the above

Q. Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

Consider the function f(x) = |sinx| + |cosx|, x ∈ R.

फलन f(x) = |sinx| + |cosx|, x ∈ R पर विचार कीजिए।

Q. Value of $f^{\prime }\left(\frac{\pi }{4}\right)$ is

प्रश्न - $f^{\prime }\left(\frac{\pi }{4}\right)$ का मान है

1. $-\sqrt{2}$
2. $\sqrt{2}$
3. 0
4. $\frac{1}{\sqrt{2}}$

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{x^2-4x+3}{x^2-1},& forx\ne 1\\ 2,& forx=1\end{array}$ , then

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

Q. Examine the continuity of the function
$f\left(x\right)=\left\{\begin{array}{ll}3x-2,& x\le 0\\ x+1,& x>0\end{array}\mathrm{at}x=0\right.$
Also sketch the graph of this function.

Q. The function $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{1}{4^x-1},& x\ne 0\end{array}\\ \begin{array}{cc}0,& x=0\end{array}\end{array}$ is continuous

1. everywhere except at $x=0$ and $x=\infty$
2. nowhere
3. everywhere
4. everywhere expect at $x=0$

Q. Given that f(x) = $\{\begin{array}{cc}\frac{x.g\left(x\right)}{|x|},& x\ne 0\\ 0,& x=0\end{array}$ g(0) = 0 = g'(0), then f'(0) equals

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

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{x-|x|}{x},& whenx\ne 0\\ 2,& whenx=0\end{array}$ , then

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

Q. Prove that the Greatest Integer Function $f:R\to R$ given by $f\left(x\right)=\left[x\right]$, is neither one-one nor onto, where $\left[x\right]$ denotes the greatest integer less that or equal to $x$