Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 15

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Which one of the following statements is not correct?

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Important Questions on Continuity and Differentiability

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 16

For

f (x) = [\begin{matrix} (x - e) 2^{- 2^{\frac{1}{e - x}}}, & x \neq e \\ 0, & x = e \end{matrix},

which of the following is/are correct ?

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 17

Let

[x]

be the greatest integer function

f (x) = \frac{\sin \frac{1}{4} π [x]}{[x]}

is

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 18

f (x) = {(\sin^{- 1} x)}^{2} \cdot \cos (\frac{1}{x})

if

x \neq 0, f (0) = 0

,

f (x)

is

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 19

f (x) = 1 + [\cos x]

in

0 < x \leq π / 2

, where

[.]

denotes greatest integer function then

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 20

If

f (x) = [x \sin π x],

then

f

is (where

[\cdot]

denotes greatest integer function)

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 21

Let $f$ be a differentiable function on the open interval $(a, b)$ . Which of the following statements must be true?

(i) $f$ is continuous on the closed interval $[a, b]$

(ii) $f$ is bounded on the open interval $(a, b)$

(iii) If $a < a_{1} < b_{1} < b$ and $f (a_{1}) < 0 < f (b_{1})$ , then there is a number $c$ such that $a_{1} < c < b_{1}$ and $f (c) = 0$

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 22

Let

f (x) = [\begin{matrix} g (x) \cdot \cos \frac{1}{x} & if x \neq 0 \\ 0 & if x = 0 \end{matrix}

where

g (x)

is an even differentiable at

x = 0

, passing through the origin. Then

f (0)

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 23

Let

f (x) = \cos x & H (x) = [\begin{matrix} Min [\frac{f (t)}{0} \leq t \leq x] & for 0 \leq x \leq \frac{π}{2} \\ \frac{π}{2} - x & for \frac{π}{2} < x \leq 3 \end{matrix}

, then