Class 12

Math

Calculus

Continuity and Differentiability

$f(x)⎩⎨⎧ 2x,ifx<00,if0≤x≤14x,ifx>1 $

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If $f(x)=max{x_{2}+2ax+1,b}$ has two points of non-differentiability, then prove that $a_{2}>1−b$

If $y=x_{(logx)_{log(logx˙)}},thendxdy is$ (a)$xy (1nx_{∞x−1})+21nx1n(1nx))$ (b)$xy (gx)_{log(logx)}(2g(gx)+1)$ (c)$x1nxy [(1nx)_{2}+21n(1nx)]$ (d)$xy gxgy [2g(gx)+1]$

Discuss the continuity of $f(x)=sgn(x_{3}−x)$

Find the points of discontinuity of the function: $f(x)=2sinx−11 $

Let $f(x)={8_{x1},x<0a[x],a∈R−{0},x≥0,$ (where [.] denotes the greatest integer function). Then (a) $f(x)$ is Continuous only at a finite number of points (b)Discontinuous at a finite number of points. (c)Discontinuous at an infinite number of points. (d)Discontinuous at $x=0$

If $y=t_{2}+t−21 ,wheret=x−11 ,$ then find the number of points where $f(x)$ is discontinuous.

Let $2f(x+y)−f(x) =2f(y)−a +xy$ for all real $xandy˙$ If $f(x)$ is differentiable and $f_{prime}(0)$ exists for all real permissible value of $a$ and is equal to $5a−1−a_{2} ˙$ Then $f(x)$ is positive for all real $x$ $f(x)$ is negative for all real $x$ $f(x)=0$ has real roots Nothing can be said about the sign of $f(x)$

Let $f(x)={x_{2}g(1+x)_{1+x}−x }˙$ Then find the value of $f(0)$ so that the function $f$ is continuous at $x=0.$