Embibe Experts Solutions for Chapter: Continuity and Differentiability, Exercise 1: Exercise

If $\left|x\right|<1,$ then $\frac{d}{dx}\left({\cos }^{-1}x-{\sin }^{-1}x\right)=$

If $y={\sin }^{-1}\left(\cos x\right)+{\cos }^{-1}\left(\sin x\right)$, then $\frac{dy}{dx}=$

Assertion : lf $f\left(x\right)=\left\{\begin{array}{ll}x+1,& \mathrm{if} x\le 1\\ 3-x^2,& \mathrm{if} x>1\end{array}\right.$, is continuous at $x=1$
Reason : A function $f \left(x\right) \mathrm{at} x =a$ is continuous then,  $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=f\left(a\right)$

Choose the correct option from given options

Show that the function $f\left(x\right)={\left[\cos \left(x^{10}+1\right)\right]}^{\frac{1}{3}}, x\in R$ is a continuous function.

Check the continuity of the function $f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{2}\left(x^2-4\right)& \text{ if }0<x<2\\ 0& \text{ if }x=2\\ 2-8x^{-3}& \text{if }x>2\end{array}\right.$ at $2$.

At $x=2, f\left(x\right)=\left[x\right]$, where $[ ]$ is a greatest integer function, is

The function $f\left(x\right)=\left|x\right|$ is

Is the function $f$, defined by $f\left(x\right)=\left\{\begin{array}{ll}{x}^2& \text{ if }x\le 1\\ x& \text{ if }x>1\end{array},\right.$ continuous on $R$?