List of top Mathematics Questions on Continuity and differentiability asked in AP EAPCET

The set of all the points at which \[ f(x)=|2-|x|| \] is continuous but not differentiable is:

If \(f(x)=\frac{\cos^{4}x-1}{x^{2}}\), \(x\ne0\) and \(f(0)=2\) is a real valued function, then:

If \[ f(x)= \begin{cases} \dfrac{x^{2}-4}{\sqrt{2-x}}, & x<2\\ a, & x=2\\ \log(x-2), & x>2 \end{cases} \] is a real valued function, then:

If the function \[ f(x)=\begin{cases}\frac{e^{b(x-1)^{2}}-1}{\sqrt{x^{2}-1}} & , \text{for } x>1 \\ \sqrt{2} & , \text{for } x=1 \\ \log\left(\frac{1+bx}{1-bx}\right)\frac{1}{\sin^{2}x} & , \text{for } 0<x<1 \end{cases} \] is continuous at \( x=1 \), then \( \lim_{x \rightarrow 2} \frac{x^{2}-5x+6}{x-2} = \)

If \( f(x)=|x-2|(3^{4|x|}-1) \) is a real valued function, then the set of points at which f is not differentiable, is

Let $[y]$ represent the greatest integer less than or equal to $y$. Then the set of all $x$ at which $f(x)=\cos^{-1}[4x+3]$ is differentiable is

If $y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + \dots \infty}$, then $(2y - 1)\frac{dy}{dx} =$

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

If $y = e^{a \sin^{-1} x}$, then $(1 - x^2) y_2 - x y_1 =$

Let \[ f(x)= \begin{cases} 4, & -\infty\lt x\lt -\sqrt5 \\ x^2-1, & -\sqrt5\leq x\leq \sqrt5 \\ 4, & \sqrt5\lt x\lt \infty \end{cases} \] If \(k\) is the number of points where \(f(x)\) is not differentiable, then \(k-2=\)

Let \[ f(x)= \begin{cases} |x|, & -\infty\lt x\lt 2 \\ |2x-4|, & 2\leq x\leq 20 \end{cases} \] \(x=a\) is a point where \(f(x)\) is continuous but not differentiable and \(x=b\) is a point where \(f(x)\) is not differentiable \((a\neq b)\). Then \(a+b=\)

\(f:\mathbb{R}\to\mathbb{R}\) is a function such that \[ |f(x)-f(y)|\leq \frac{1}{2}|x-y| \quad \forall x,y\in \mathbb{R} \] and \[ f'(x)\geq \frac{1}{2}\quad \forall x\in \mathbb{R},\quad f(1)=\frac{1}{2} \] Then the number of points of intersection of the curve \(y=f(x)\) and the curve \[ y=x^2-2x-5 \] is:

If \[ f(x)= \begin{cases} \dfrac{x^2\log(\cos x)}{\log(1+x)}, & x\neq 0 \\ 0, & x=0 \end{cases} \] then at \(x=0\), \(f(x)\) is

If \[ y=\sin\left(2\tan^{-1}\sqrt{\frac{1-x}{1+x}}\right), \quad x=\cos 2\theta, \] then \[ \frac{dy}{dx}= \]

If \[ f(x)=|x^2-3x+2|, \] then \[ \frac{df}{dx}= \]