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Continuity And Differentiability - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • 42 Questions around this concept.

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QuestionMEDIUM

Number of points where \mathrm{f(x)=x^{2}-\left|x^{2}-1\right|+2|| x|-1|+2|x|-7}  is non-differentiable is

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 If \mathrm{f(x)=x, x \leq 1}, and \mathrm{f(x)=x^2+b x+c, x>1}, and \mathrm{f^{\prime}(x)} exists finitely for all \mathrm{x \in R} then.
 

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 \begin{aligned}\text{if } \mathrm{ f(x)}&=\mathrm{e^x, x<2}\\ &\mathrm{a+b x, x \geq 2} \end{aligned} is differentiable for all \mathrm{x \in R} then.
 

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If \mathrm{f(x)=\cos ^{-1}(\cos x)}  then \mathrm{f(x)}  is.
 

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Let \mathrm{f(x)=x |x| }. The set of points where \mathrm{f(x) } is twice differentiable, is:

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\mathrm{ If \, \, f(x+y)=f(x)+f(y)+|x| y+x y^2 \forall x, y \in R\, \, and\, \, f(0)=0, then : }

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The set of all points, where the function \mathrm{ f(x)=\frac{x}{1+|x|} } is differentiable is :

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The function given by  \mathrm{y=|| x|-1|}  is differentiable for all real numbers except the points :

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The domain of the derivative of the function
\mathrm{f(x)=\left\{\begin{array}{c} \tan ^{-1} x ; \text { if }|x| \leq 1 \\ \frac{1}{2}(|x|-1) ; \text { if }|x|>1 \text { is } \end{array}\right.}

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Which of the following functions is not differentiable in the domain \mathrm{[-1,1]} ?

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Concepts Covered - 1

Differentiability in an Interval and Theorems of Differentiability

Differentiability in an Interval

A)  A function f(x) is differentiable in an open in interval (a,b) if it is differentiable at every point on the opern interval (a,b).

B) A function y = f (x) is said to be differentiable in the closed interval [a, b].

  1. If f (x) is differentiable at every point on the open interval (a, b). And,

  2. It is differentiable from the right at “a” and from the left at “b”.

            (In other words, \lim_{x\rightarrow a^+}\frac{f(x)-f(a)}{x-a}\;\;\text{and}\;\;\lim_{x\rightarrow b^-}\frac{f(x)-f(b)}{x-b} both exists), then f(x) is said to be differentiable in [a, b]

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Differentiability in an Interval and Theorems of Differentiability

Study other Related Concepts

Continuity And Differentiability Current Topic

Limit
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

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Reference Books

Differentiability in an Interval and Theorems of Differentiability

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 4.18

Line : 32

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