Embibe Experts Solutions for Chapter: Continuity and Differentiability, Exercise 1: JEE Main - 22 July 2021 Shift 1

If for $p\ne q\ne 0$, then function $f\left(x\right)=\frac{\sqrt[7]{p\left(729+x\right)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then

Let $f:\left[0,1\right]\to R$ be a twice differentiable function in $\left(0,1\right)$ such that $f\left(0\right)=3$ and $f\left(1\right)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $\left(0,1\right)$, then the least number of points $x\in \left(0,1\right)$, at which $f^{"}\left(x\right)=0$, is

The function $f:R\to R$ defined by $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{\cos \left(2\pi x\right)-x^{2n}\sin \left(x-1\right)}{1+x^{2n+1}-x^{2n}}$ is continuous for all $x$ in

The number of points, where the function $f:R\to R, f\left(x\right)=\left|x-1\right|\cos \left|x-2\right|\sin \left|x-1\right|+\left(x-3\right)\left|x^2-5x+4\right|$, is NOT differentiable, is

Let the function $f\left(x\right)=\left\{\begin{array}{cc}\frac{{\log }_e\left(1+5x\right)-{\log }_e\left(1+\alpha x\right)}{x}& \mathrm{if} x\ne 0\\ 10& \mathrm{if} x=0\end{array}\right.$ be continuous at $x=0$. Then $\alpha$ is equal to

If $\left[t\right]$ denotes the greatest integer $\le t$, then number of points, at which the function $f\left(x\right)=4\left|2x+3\right|+$ $9\left[x+\frac{1}{2}\right]-12\left[x+20\right]$ is not differentiable in the open interval $\left(-20,20\right)$, is ______.

Let $f\left(x\right)=\left\{\begin{array}{l}{x}^2\sin \left(\frac{1}{x}\right); x\ne 0\\ 0; x=0\end{array}\right.$, then at $x=0$

If the function $f\left(x\right)=\left\{\begin{array}{cc}\left(1+|\cos x|\right)\frac{\lambda }{|\cos x|},& 0<x<\frac{\pi }{2}\\ \mu ,& x=\frac{\pi }{2}\\ e^{\frac{\cot 6x}{\cot 4x}},& \frac{\pi }{2}<x<\pi \end{array}\right.$is continuous at $x=\frac{\pi }{2}$, then $9\lambda +6 {\log }_c\mu +{\mu }^6-e^{6\lambda }$ is equal to