Continuity and Differentiability Test

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Continuity and Differentiability Test - 41

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Question 1
1 / -0

Given that $$f(x)$$ is a differentiable function of $$x$$ and that $$f\left( x \right).f\left( y \right) = f\left( x \right) + f\left( y \right) + f\left( {xy} \right) - 2$$ and that $$f\left( 2 \right) = 5.$$ Then $$f'\left( 3 \right)$$ is equal to

A
$$6$$

B
$$24$$

C
$$15$$

D
$$19$$

Solution
Question 2
1 / -0

If $$y^{y^{y^{....{^\infty}}}} = \log_e(x+\log_e(x+....))$$, then $$\dfrac{dy}{dx}$$ at $$(x= e^2-2, y= \sqrt2)$$ is

A
$$\dfrac{\log\left(\dfrac{e}{2}\right)}{2\sqrt2(e^2-1)}$$

B
$$\dfrac{\log2}{2\sqrt2(e^2-1)}$$

C
$$\dfrac{\sqrt2 \log\dfrac{e}{2}}{(e^2-1)}$$

D
None of these

Solution
Question 3
1 / -0

Let $$y=x^{x^x}$$, then differentiate $$y$$ w.r.t $$x$$.

A
$$x^{x^x}\left(\dfrac{1}{x}+\log x+(\log x)^2\right)$$

B
$$x^{x^x}(x^x)\left(\dfrac{1}{x}+\log x-(\log x)^2\right)$$

C
$$x^{x^x}(x^x)\left(\dfrac{1}{x}+\log x+(\log x)^2\right)$$

D
$$x^{x^x}(x^x)\left(\dfrac{1}{x}-\log x-(\log x)^2\right)$$

Solution

$$y=x^{x^x}$$
diffrentiate it w.r.t.$$x$$
$$\dfrac{dy}{dx}=(e^{lnx^{x^x}})^{'}$$
$$\dfrac{dy}{dx}=(e^{x^x lnx})^{'}$$
$$\dfrac{dy}{dx}=(e^{e^{x lnx}lnx)})^{'}$$
$$\dfrac{dy}{dx}=(e^{e^{x lnx}lnx)})(e^{xlnx}lnx)^{'}$$
$$\dfrac{dy}{dx}=(e^{e^{x lnx}lnx)})(e^{xlnx}(xlnx)^{'}ln x$$+$$\dfrac{e^{xlnx}}{x})$$
$$\dfrac{dy}{dx}=(e^{e^{x lnx}lnx)})(e^{xlnx}(xlnx)^{'}ln x$$+$$\dfrac{e^{xlnx}}{x})$$
$$\dfrac{dy}{dx}=(e^{e^{x lnx}lnx)})(e^{xlnx}(lnx+\dfrac{x}{x})ln x$$+$$\frac{e^{xlnx}}{x}$$
$$\dfrac{dy}{dx}=x^{x^{x}}$$$$(x^x(lnx+1)lnx+\dfrac{x^x}{x}$$
$$\dfrac{dy}{dx}=x^{x^{x}}$$$$(x^x)(\dfrac{1}{x}+lnx+(lnx)^2)$$
so option C is correct.
Question 4
1 / -0

If $$x = 2\left( {\theta + \sin \theta } \right)and\,y = 2\left( {1 - \cos \theta } \right),then\;value\;of\frac{{dy}}{{dx}}\;is\;$$

A
$$\tan \left( {\frac{\theta }{2}} \right)$$

B
$$\cot \left( {\frac{\theta }{2}} \right)$$

C
$$\sin \left( {\frac{\theta }{2}} \right)$$

D
$$\cos \left( {\frac{\theta }{2}} \right)$$
Question 5
1 / -0

If $$f\left( x \right) = x + \left| x \right| + {\kern 1pt} \,\cos \left( {\left[ {{\pi ^2}} \right]x} \right)\,\,and\,\,g\left( x \right) = \,\sin \,x\,where\,\left[ . \right]$$ denotes the greatest integer function) then :-
$$(1)$$ $$f\left( x \right) + g\left( x \right)$$ is discontinuous
$$(2)$$$$f\left( x \right) + g\left( x \right)$$ is differentiable everywhere
$$(3)$$ $$f\left( x \right) \times g\left( x \right)$$ is differentiable everywhere
$$(4)$$ $$f\left( x \right) \times g\left( x \right)$$ is countimuos but not diffrentiable at $$x=0$$

A
1

B
2

C
3

D
4
Question 6
1 / -0

If $$f\left( x \right) = \left\{ \begin{array}{l}\frac{{1 - \left| x \right|}}{{1 + x}},{\rm{ }}x \ne - 1\\1,{\rm{ }}x = - 1{\rm{ }}\end{array} \right.$$ then $$f\left( {\left[ {2x} \right]} \right),$$ where $$\left[ {} \right]$$ represents the greatest integer function , is

A
discontinuous at $$x = - 1$$

B
continuous at $$x = 0$$

C
continuous at $$x = \frac{1}{2}$$

D
continuous at $$x = 1$$

Solution
Question 7
1 / -0

if $$y^{2}=ax+bx+c$$, then $$ y^{3} \dfrac {d^{2}y}{dx^{2}}$$ is

A
a constant

B
a function of $$x$$ only

C
a function of $$y$$ only

D
a function of $$x$$ and $$y$$

Solution

Given $$y^2=ax+bx=x$$
differentiate w.r.t. to x
$$2y\cdot \dfrac{dy}{dx}=a+b$$ …….$$(1)$$
Again differentiate wrt to x
$$2\left(\dfrac{dy}{dx}\right)^2+2y\cdot\dfrac{d^2y}{dx^2}=0$$
$$\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}=0$$
$$y\dfrac{d^2y}{dx^2}=-\left(\dfrac{dy}{dx}\right)^2$$
Multiplying both side by $$y^2$$
$$y^3\dfrac{d^2y}{dx^2}=-y^2\left(\dfrac{dy}{dx}\right)^2$$
From equation $$(1)$$
$$y^3\dfrac{d^2y}{dx^2}=-y^2\left(\dfrac{a+b}{2y}\right)^2$$
$$y^3\dfrac{d^2y}{dx^2}=-\dfrac{(a+b)^2}{4}$$
So, $$y^3\dfrac{d^2y}{dx^2}$$ is a constant.
$$\therefore$$ Option A is correct.
Question 8
1 / -0

If $$f:\left[ {0,1} \right] \to \left[ {0,1} \right]$$ be definded by f(x) =\begin{cases} x,\quad \quad \quad \quad \quad \quad if\quad x\quad is\quad rational\quad \\ 1-x,\quad \quad \quad \quad if\quad x\quad is\quad irrational\quad \quad \quad \quad \quad \end{cases} then $$\left( {f \circ f} \right)x$$ ______________.

A
constant

B
1+x

C
x

D
None of these

Solution
Question 9
1 / -0

If $$\left( \frac { 1-x }{ 1+x } \right) =x$$ and $$g\left( x \right) =\int { f\left( x \right) } dx$$ then

A
$$g\left( x \right)$$ is continuous in domain

B
$$g\left( x \right)$$ is discontinuous st two points in its domain

C
$$\lim _{ x\rightarrow \infty }{ g\left( x \right)=-1 }$$

D
$$\int { g\left( x \right) dx=-\frac { { x }^{ 2 } }{ 2 } +\left( 2x+1 \right) \lambda n\left( \frac { 1+x }{ e } \right) +C } $$
Question 10
1 / -0

If $$f(x)=\begin{cases} \dfrac { 1-\sqrt { 2 } \sin { x } }{ \pi -4x } ,\quad \quad ifx\neq \dfrac { \pi }{ 4 } \\ a\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ,\quad \quad ifx=\dfrac { \pi }{ 4 } \end{cases}$$ is continous at $$x=\dfrac {\pi}{4}$$ then $$a=$$

A
$$4$$

B
$$2$$

C
$$1$$

D
$$1/4$$

Solution