Continuity and Differentiability Test

Result

Continuity and Differentiability Test - 42

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Question 1
1 / -0

$$f(x)= x\sin\dfrac{1}{x} , \ for x\neq 0$$
$$= 0,\ for x=0$$
Then.

A
$$f'(0^+) exit\ but \ f'(0^-)$$ does not exit

B
$$f'(0^+) \ and \ f'(0^-)$$ do not exit

C
$$f'(0^+) = f'(0^-)$$

D
none of these

Solution
Question 2
1 / -0

A function $$f$$ satisfies the relation $$f(x)=f'(x)+f"(x)+.....\infty$$ terms, where $$f(x)$$ is differentiable indefinitely, If $$f'(-1)=1$$ then $$f(-1)$$ is equal to

A
$$0$$

B
$$1$$

C
$$2$$

D
$$4$$

Solution
Question 3
1 / -0

$$f(x) = \left\{\begin{matrix}(3/x^{2})\sin 2x^{2} & if x M 0 \\\dfrac {x^{2} + 2x + c}{1 - 3x^{2}} & if\ x \geq 0, x \neq \dfrac {1}{\sqrt {3}}\\ 0 & x = 1/ \sqrt {3}\end{matrix}\right.$$ then in order that $$f$$ be continuous at $$x = 0$$, the value of $$c$$ is

A
$$2$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution
Question 4
1 / -0

Let $$g(x)$$ be a continuous function for all $$x$$, and $$f(x)=f(\alpha)+(x-\alpha).g(x)\ \forall \ x\ =\epsilon \ R$$. Then;

A
$$f(x)$$ is necessarily differentiable at $$x=\alpha$$

B
$$f(x)$$ is not necessarily differentiable at $$x=\alpha$$

C
$$f(x)$$ is not necessarily continuous at $$x=\alpha$$

D
$$None\ of\ these$$

Solution
Question 5
1 / -0

Let a function $$f: R\rightarrow R$$ be given by $$f(x+y)=f(x)f(y)$$ for all $$x, y \in R$$ and $$f(x)\neq 0$$ for any $$x_{1}$$ function $$f(x)$$ is differentiable at $$x=0$$. Find $$f(x) gi ven f(0)=1$$.

A
$$e^{x}$$

B
$$x.{f'(0)}$$

C
$$\dfrac{x^{2}}{2}f'(x)$$

D
$$None\ of\ these$$

Solution
Question 6
1 / -0

$$\dfrac { d }{ dx } \left[ { cos }^{ -1 }\left( x\sqrt { x } -\sqrt { \left( 1-x \right) \left( 1-{ x }^{ 2 } \right) } \right) \right] =$$

A
$$\dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } -\dfrac { 1 }{ 2\sqrt { { x-x }^{ 2 } } }$$

B
$$\dfrac { -1 }{ \sqrt { 1-{ x }^{ 2 } } } -\dfrac { 1 }{ 2\sqrt { { x-x }^{ 2 } } }$$

C
$$\dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } +\dfrac { 1 }{ 2\sqrt { { x-x }^{ 2 } } }$$

D
$$\dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$$

Solution
Question 7
1 / -0

If f is twice differentianle such that f"$$\left( x \right) = - f\left( x \right),f'\left( x \right) = g\left( x \right),h'\left( x \right) = {\left( {f\left( x \right)} \right)^2} + {\left( {g\left( x \right)} \right)^2}$$ and $$h\left( 0 \right) = 2,h\left( 1 \right) = 4$$,then equation of $$h\left( x \right)$$ represents ?

A
curve of degree $$2$$

B
curve passing through origin

C
a straight line with slope $$2$$

D
a straight line with slope $$-2$$

Solution
Question 8
1 / -0

$$\frac{d}{{dx}}\left[ {a{{\tan }^{ - 1}}x + b\log \left( {\frac{{x - 1}}{{x + 1}}} \right)} \right] = \frac{1}{{{x^4} - 1}} \Rightarrow a - 2b = $$

A
$$1$$

B
$$-1$$

C
$$0$$

D
$$2$$
Question 9
1 / -0

Let $$f$$ be the function on $$[0,\,1]$$ given by $$f\left( x \right) = x\sin \frac{\pi }{x}\,for\,x \ne 0$$ and $$f(0)=0$$. Then

A
Continuous but bounded

B
Differentiable

C
Continuous but not bounded

D
Continuous and bounded

Solution
Question 10
1 / -0

If $$f(x)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}$$, then $$f(x)$$ is differentiable on

A
$$\left(-\infty,\infty\right)$$

B
$$\left[2,\infty\right)-\left\{4\right\}$$

C
$$\left[2,\infty\right)$$

D
$$\left(0,\infty\right)$$

Solution