Theorems for Differentiability

Q. Which of the following functions are identical?

1. $f\left(x\right)=\ln x^2\mathrm{and}g\left(x\right)=2\ln x$
2. $f\left(x\right)={\log }_{x}e\mathrm{and}g\left(x\right)=\frac{1}{{\log }_{e}x}$
3. $f\left(x\right)=\sin \left({\cos }^{-1}x\right)\mathrm{and}g\left(x\right)=\cos \left({\sin }^{-1}x\right)$
4. None of these

Q. $\underset{x\to 1}{lim}\frac{x\sin \left\{x\right\}}{x-1},$ where {x} denotes the fractional part of x, is equal to

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

Q. The sum of the solutions of the equation
$2{\sin }^{-1}\left(\sqrt{x^2+x+1}\right)+{\cos }^{-1}\left(\sqrt{x^2+x}\right)=\frac{3\pi }{2}\mathrm{is}$

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

Q. Let $a,bϵR$ and $f:R\to R$ be defined by $f\left(x\right)=acos(|x^3-x|)+b|x|sin(|x^3+x|)$.
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. Differentiable at x=0 if a =1 b=0
4. Differentiable at x=1 if a =1 b=1

Q. The range of $f\left(x\right)=\left[x\right]-x.$ Where $\left[x\right]$ is the greatest integer function.

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

Q. If a function $f\left(x\right)$ is defined as $f\left(x\right)=\{\begin{array}{cc}-x,& x<0\\ x^2,& 0\le x\le 1\\ x^2-x+1,& x>1\end{array},$
then which among the following option is correct

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

Q. Let g(x) = x.f(x), where f(x) = $f\left(x\right)=\{\begin{array}{cc}xsin\frac{1}{x},& x\ne 0\\ 0,& x=0.\end{array}$ at x = 0

1. g is differentiable but g' is not continuous
2. g is differentiable while f is not

3. Both f and g are differentiable

4. g is differentiable and g' is continuous

Q. Let $f(x+y)=f\left(x\right)+f\left(y\right)$ and $f\left(x\right)=x^{2}g\left(x\right)$ for all $x,yϵR$, where $g\left(x\right)$ is continuous function. Then $f^{\prime }\left(x\right)$ is equal to

1. g'(x)
2. g(0)
3. g(0) + g'(x)
4. \N