If f ( x ) = x x - x + 1 , then indicate the correct alternative(s):

f ( x ) is differentiable at x = 0 .

f ( x ) is continuous but not differentiable at x = 0 .

f x is not differentiable at x = 0 .

If f ( x ) = x x - x + 1 , then indicate the correct alternative(s):

f ( x ) is differentiable at x = 0 .

If f x = x 3 e 1 x + 4 2 - e 1 x , x ≠ 0 0 , x = 0 , then f x is

continuous but not differentiable at x = 0 .

continuous as well differentiable at x = 0 .

neither differentiable at x = 0 nor continuous at x = 0 .

If f x = x x + 1 - x , be a real valued function, then

f ( x ) is differentiable at x = 0 .

f ( x ) is continuous, but f ' ( 0 ) does not exist.

f ( x ) is not continuous at x = 0 .

f ( x ) is not differentiable at x = 0 .

Given f x = log a a [ x ] + [ - x ] x a 2 x + - x x - 5 3 + a 1 | x | for x ≠ 0 0 for x = 0 ; a > 1 where [.] represents the integral part function, then

f is continuous & differentiable at x = 0

f is continuous but not differentiable at x = 0

the differentiability of function f at x = 0 depends on the value of a

function f is continuous & differentiable at x = 0 and for a = e only.

If f x = x 2 - 1 x 2 + 1 , 0 < x ≤ 2 1 4 x 3 - x 2 , 2 < x ≤ 3 9 4 x - 4 + 2 - x , 3 < x < 4 , then

f ( x ) is non-differentiable at x = 2 & x = 3 .

f ( x ) is differentiable at x = 2 & x = 3 .

f ( x ) is differentiable at x = 3 but not at x = 2 .

f ( x ) is differentiable at x = 2 but not at x = 3 .

Let f ( x ) be defined in [ - 2 , 2 ] by f x = max ( 4 - x 2 , 1 + x 2 ) , - 2 ≤ x ≤ 0 min ( 4 - x 2 , 1 + x 2 ) , 0 < x ≤ 2 , then f x

is not differentiable at more than one point.

is not continuous at more than one point.

is not differentiable only at one point.

Let f x = x - x 2 and g x = max ( f t , 0 ≤ t ≤ x ) , 0 ≤ x ≤ 1 sinπ x , x > 1 , then in the interval [ 0 , ∞ )

g ( x ) is everywhere differentiable except at x = 1

g ( x ) is everywhere continuous except at two points

g ( x ) is everywhere differentiable except at two points

Let f : [ a , b ] → [ 1 , ∞ ) be a continuous function and let g : R → R be defined as g x = 0 i f x < 0 ∫ a x f t d t i f a ≤ x ≤ b ∫ a b f t d t i f x > b . Then :

g x is continuous but not differentiable at b

g x is continuous and differentiable at either a or b but not both

For every twice differentiable function f : ℝ → - 2 , 2 with f 0 2 + f ′ 0 2 = 85 , which of the following statement(s) is (are) TRUE?

There exists α ∈ - 4 , 4 such that f α + f ′ ′ α = 0 a n d f ′ α ≠ 0

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 56

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If $f (x) = x (\sqrt{x} - \sqrt{x + 1})$ , then indicate the correct alternative(s):

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Important Questions on Continuity and Differentiability

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 57

If

f (x) = \{\begin{array}{cl} \frac{x (3 e^{\frac{1}{x}} + 4)}{2 - e^{\frac{1}{x}}}, & x \neq 0 \\ 0, & x = 0 \end{array}

, then

f (x)

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 58

If

f (x) = \frac{x}{\sqrt{x + 1} - \sqrt{x}}

, be a real valued function, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 61

Given $f (x) = \{\begin{matrix} \log_{a} {(a |[x] + [- x]|)}^{x} (\frac{a^{\frac{2}{(\frac{[x] + [- x]}{|x|})} - 5}}{3 + a^{\frac{1}{| x |}}}) & for & x \neq 0 \\ 0 & for & x = 0 \end{matrix}$ ; $a > 1$ where [.] represents the integral part function, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 62

If $f (x) = \{\begin{cases} \frac{x^{2} - 1}{x^{2} + 1} & , 0 < x \leq 2 \\ \frac{1}{4} (x^{3} - x^{2}) & , 2 < x \leq 3 \\ \frac{9}{4} (|x - 4| + |2 - x|) & , 3 < x < 4 \end{cases}, then$

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 64

If

f (x)

is differentiable everywhere, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 65

Let $f (x)$ be defined in $[- 2, 2]$ by $f (x) = \{\begin{cases} \max (\sqrt{4 - x^{2}}, & \sqrt{1 + x^{2}}), - 2 \leq x \leq 0 \\ \min (\sqrt{4 - x^{2}}, & \sqrt{1 + x^{2}}), 0 < x \leq 2 \end{cases}$ , then $f (x)$

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 67

Let

f (x) = x - x^{2}

and

g (x) = \{\begin{matrix} \begin{matrix} \max (f (t), & 0 \leq t \leq x), \end{matrix} & 0 \leq x \leq 1 \\ sinπ x, & x > 1 \end{matrix}

, then in the interval

[0, \infty)

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 68

Consider the following statements:
$S_{1} :$ Number of points where $f (x) = |x sgn (1 - x^{2})|$ is non-differentiable is $3$
$S_{2} :$ Defined $f (x) = \{\begin{cases} a \sin (\frac{π}{2} (x + 1)) & , x \leq 0 \\ \frac{\tan x - \sin x}{x^{3}} & , x > 0 \end{cases}$ , in order that, $f (x)$ is continuous, $' a'$ should be equal to $\frac{1}{2}$

$S_{3} :$ The set of all points, where the function $\sqrt[3]{x^{2} | x |}$ is differentiable is $(- \infty, 0) \cup (0, \infty)$
$S_{4} :$ Number of points where $f (x) = \frac{1}{\sin^{- 1} (\sin x)}$ is non-differentiable in the interval $(0, 3 π)$ is $3 .$ State, in order, whether $S_{1}, S_{2}, S_{3}, S_{4}$ are true or false