Continuity and Differentiability Test

Result

Continuity and Differentiability Test - 45

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Question 1
1 / -0

The derivate of $$\tan^{-1}[\dfrac {\sin x}{1+\cos x}]$$ with respect to $$\tan^{-1}[\dfrac {\cos x}{1+\sin x}]$$ is

A
$$2$$

B
$$-1$$

C
$$0$$

D
$$-2$$

Solution
Question 2
1 / -0

$$S$$ be the set in which function defined by $$f(x)=\sin |x|- |x|+2(x-\pi)\cos |x|$$ is differtentiable then no of element in $$S$$ is

A
$$2$$

B
$$3$$

C
$$4$$

D
$$\phi$$

Solution
Question 3
1 / -0

If $$y = \sin ^ { - 1 } \dfrac { 2 x } { 1 + x ^ { 2 } } ,$$ then which of the following is not correct ?

A
$$\dfrac { d y } { d x } = \dfrac { 2 } { 1 + x ^ { 2 } }$$ for $$| x | < 1$$

B
$$\dfrac { d y } { d x } = \dfrac { - 2 } { 1 + x ^ { 2 } }$$ for $$| x | > 1$$

C
$$\dfrac { d y } { d x } = 2$$ for $$x = - 1$$

D
$$\dfrac { d y } { d x }$$ does not exist at $$| x | = 1$$

Solution
Question 4
1 / -0

If $$f(x)$$ is differentiable function and $$f(1) = \sin 1, f(2) = \sin 4, f(3) = \sin 9$$, then the minimum number of distinct solutions of equation $$f'(x) = 2x \,\cos \,x^2$$ in $$(1, 3)$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 5
1 / -0

If $$y = {\cot ^{ - 1}}\left[ {\dfrac{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} }}{{\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right]\;then\;\dfrac{{dy}}{{dx}}\;is\;equal\;to\;$$

A
$$\dfrac{1}{2}$$

B
$$\dfrac{2}{3}$$

C
$$3$$

D
2
Question 6
1 / -0

If $$f\left(x\right)$$ is a differentiable function in the interval $$\left(0,\infty\right)$$ such that $$f\left(1\right)=1$$ and $$\lim_{t\rightarrow x}\dfrac{t^{2}f\left(x\right)-x^{2}f\left(t\right)}{t-x}=1$$, for each $$x>0$$, then $$f\left(\dfrac{3}{2}\right)$$ is equal to:

A
$$\dfrac{25}{9}$$

B
$$\dfrac{13}{6}$$

C
$$\dfrac{23}{18}$$

D
$$\dfrac{31}{18}$$
Question 7
1 / -0

Let $$f\left( x \right)=x\left| x \right| ,g\left( x \right)=sinx$$ and $$h\left( x \right) =\left( gof \right) \left( x \right) .$$ Then

A
$$h'\left( x \right) $$is differentiable at x=0

B
$$h'\left( x \right) $$ is continuous at x=0 but is not differentiable at x=0

C
$$h'\left( x \right) $$ is differentiable at x=0 but $$h'\left( x \right) $$ differentiable is not continuous at x=0

D
$$h'\left( x \right) $$ is not differentiable at x=0

Solution
Question 8
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Let f be a differentiable function satisfying the condition $$f\left( \dfrac { x }{ y } \right) =\dfrac { f\left( x \right) }{ f\left( y \right) } ,$$ for all x, y$$\left( \neq 0 \right) \epsilon R$$ and f(y)$$\neq $$0. If f'(1)=2, then f'(x) is equal to

A
2 f(x)

B
$$\dfrac { f\left( x \right) }{ x } $$

C
2x f(x)

D
$$\dfrac { 2f\left( x \right) }{ x } $$

Solution
Question 9
1 / -0

Directions For Questions

Consider a differentiable $$f:R\rightarrow R$$ for which $$f'(0)=\log 2$$ and $$f(x+y)=2^{x} f(y)+4^{y} f(x) \forall x, y \in R$$

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The value of $$f(4)$$ is

A
$$160$$

B
$$240$$

C
$$200$$

D
$$None\ of\ these $$

Solution
Question 10
1 / -0

Let f be a differentiable function satisfying the relation f(xy) = xf(y) - 2xy+yf(x) (where x,y > 0) and f(1) = 3, then

A
f(x) =x $$\ell nx+3x-\frac { { x }^{ 2 } }{ 2 } $$

B
$$f(x)\quad =\quad x\ell nx$$

C
$$x={ e }^{ -3 }$$ is the abscissa of the point of inflection of f(x)

D
the equation f(x) = k has two solution if x $$\in \quad (-{ e }^{ -3 },0)$$

Solution