JEE Main · 2005 · 08 May 2005 · Offline · Continuity and Differentiability

More JEE Main questions on Continuity and Differentiability

1. Let $[\bullet ]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2}x,x^2\right\}$. Let $S=\{x\in (-2,2):$ the function $\mathrm{g}(\mathrm{x})=|\mathrm{x}|\left[{\mathrm{x}}^2\right]$ is discontinuous at x$\}$.
   
   Then ${\sum }_{x\in S}f(x)$ equa …
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2. Let $f(x)=\{\begin{array}{ll}\frac{a{x}^2+2ax+3}{4x^2+4x-3},& x\ne -\frac{3}{2},\frac{1}{2}\\ b& x=-\frac{3}{2},\frac{1}{2}\end{array}$
   be continuous at $x=-\frac{3}{2}$. If $\mathrm{fof}(x)=\frac{7}{5}$, then $x$ is equal to :
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3. If $f(x)=\{\begin{array}{cl}\frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x}& ,x\ne 0\\ b& ,x=0\end{array}$
   is continuous at $\mathrm{x}=0$, then $\mathrm{a}+\mathrm{b}$ is equal to :
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4. If the function $\mathrm{f}(\mathrm{x})=\frac{e^x\left(e^{\tan x-x}-1\right)+{\log }_e(\sec x+\tan x)-x}{\tan x-x}$ is Continuous at $\mathrm{x}=0$, then the value of $\mathrm{f}(0)$ is equal to
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5. Let $\alpha ,\beta \in \mathbb{R}$ be such that the function
   $f(\mathrm{x})=\{\begin{array}{ll}2\alpha \left({\mathrm{x}}^2-2\right)+2\beta \mathrm{x}& ,\mathrm{x}<1\\ (\alpha +3)\mathrm{x}+(\alpha -\beta )& ,\mathrm{x}\ge 1\end{array}$
   be differentiable at all …
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6. Consider the following three statements for the function $\mathrm{f}:(0,\infty )\to \mathbb{R}$ defined by
   $\mathrm{f}(\mathrm{x})=\left|{\log }_{\mathrm{e}}\mathrm{x}\right|-|\mathrm{x}-1|:$
   (I) f is differentiable at all $\mathrm{x}>0$.
   (II) f is increasing in …
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