Continuity and Differentiability Test

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Continuity and Differentiability Test - 30

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Weekly Quiz Competition

Question 1
1 / -0

If Rolle's therorem holds for $$f(x)=x\left( { x }^{ 2 }+ax+b \right) +2atx=\frac { 1 }{ 2 } $$in the interval (-1 , 1), then which of the following (S) is/are corect?

A
a + b =$$-\frac { 3 }{ 4 } $$

B
a + b = $$-\frac { 5 }{ 4 } $$

C
ab = $$\frac { 1 }{ 4 } $$

D
ab = $$-\frac { 1 }{ 4 } $$

Solution
Question 2
1 / -0

Let $$f\left( x \right)$$ be non-constant differentiable function for all real $$x$$ and $$f\left( x \right) =f\left( 1-x \right) $$. Then Rolle's theorem is not applicable for $$f\left( x \right)$$ on

A
$$[0,1]$$

B
$$[-1,2]$$

C
$$[-2,3]$$

D
$$\left[ 0,\dfrac { 2 }{ 3 } \right] $$

Solution
Question 3
1 / -0

in $$\left[ a,b \right] $$ and differentiable in (a,b) then the value of 'c' for the pair of functions
$$f\left( x \right) \sqrt { x, } \phi \left( x \right) =\dfrac { 1 }{ \sqrt { x } } $$ is

A
$$\sqrt { a } $$

B
$$\sqrt { b } $$

C
$$\sqrt { ab } $$

D
$$-\sqrt { ab } $$
Question 4
1 / -0

Let f(x) be a polynomial in x. Then the second derivation of f$$\left( { e }^{ x } \right) $$, is:

A
$$f"\left( { e }^{ x } \right) .{ e }^{ x }+f'\left( { e }^{ x } \right) $$

B
$$f"\left( { e }^{ x } \right) .{ e }^{ x }+f'\left( { e }^{ x } \right) .{ e }^{ 2x }$$

C
$$f"\left( { e }^{ x } \right) { e }^{ 2x }$$

D
$$f"\left( { e }^{ x } \right) { e }^{ 2x }+f'\left( { e }^{ x } \right) .{ e }^{ x }$$

Solution

We have,

$$f\left( x \right)$$ is a polynomial.

Then,

The second order of $$f\left( {{e}^{x}} \right)=?$$

Find that,

$$\dfrac{d}{dx}\left[ \dfrac{d}{dx}f\left( {{e}^{x}} \right) \right]=?$$

So,

We know that,

$$ \dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}} $$

$$ So, $$

$$ \dfrac{d}{dx}\left[ f\left( g\left( x \right) \right) \right]=g'\left( x \right)f'\left( g\left( x \right) \right) $$

Now. Using formula,

$$\dfrac{d}{dx}\left( I.II \right)=I.\dfrac{d}{dx}II+II.\dfrac{d}{dx}I$$

So,

$$\dfrac{d}{dx}\left[ f\left( {{e}^{x}} \right) \right]={{e}^{x}}.f'\left( {{e}^{x}} \right)\,\,\,......\,\,\left( 1 \right)$$

Again differentiating and we get,

$$ \dfrac{d}{dx}\left[ \dfrac{d}{dx}f\left( {{e}^{x}} \right) \right]=\dfrac{d}{dx}\left[ f'\left( {{e}^{x}} \right).{{e}^{x}} \right]=f'\left( {{e}^{x}} \right){{e}^{x}}+{{e}^{x}}\left[ {{e}^{x}}f''\left( {{e}^{x}} \right) \right] $$

$$ =\dfrac{d}{dx}\left[ \dfrac{d}{dx}f\left( {{e}^{x}} \right) \right]={{e}^{x}}f'\left( {{e}^{x}} \right)+{{e}^{2x}}f''\left( {{e}^{x}} \right) $$

Hence, this is the answer.
Question 5
1 / -0

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

A
$$2f(x)$$

B
$$f(x)/x$$

C
$$2xf(x)$$

D
$$2f(x)/x$$

Solution
Question 6
1 / -0

Solve $$ x ^ { 4 } \left( \dfrac { d y } { d x } \right) ^ { 2 } - x \dfrac { d y } { d x } - y = 0$$

A
$$y = x C ^ { 2 } - \dfrac { C } { x }$$

B
$$y = x ^ { 2 } C ^ { 2 } - \dfrac { C } { x }$$

C
$$y = x ^ { 2 } C - \dfrac { C } { x }$$

D
None of these
Question 7
1 / -0

If
$$f\left( x \right) =\dfrac { \log { \left( 1+ax \right) -\log { \left( 1-bx \right) } } }{ \begin{matrix} x \\ =-c \end{matrix} } ,\begin{matrix} \\ \\ x\neq 0 \\ \\x=0 \end{matrix}$$ and is continuous at $$x=0$$

then the line $$ax+by+c=0$$ passes though the point

A
$$(1,-1)$$

B
$$(-1,1)$$

C
$$(-1,-1)$$

D
$$(1,1)$$

Solution
Question 8
1 / -0

$$\dfrac{d}{dx}$$ { $$\cot^{-1} \dfrac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}$$ } =

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

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

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

D
None of these

Solution
Question 9
1 / -0

If $$f(x)$$ is a four times differentiable even function, then $$\int_{-3}^{3}(x^{3}f(x)+xf''''(x)+2)dx$$ is equal to

A
$$12f(x)+f''(x))$$

B
$$12f''(x)$$

C
$$12$$

D
$$6$$

Solution
Question 10
1 / -0

Let f, g and h are function differentiable on some open interval around $$0$$ and satisfy the equations $$f'=2f^2gh+\dfrac{1}{gh}, f(0)=1, g'=fg^2h+\dfrac{4}{fh}, g(0)=1$$ and $$h'=3fgh^2+\dfrac{1}{fg}, h(0)=1$$. The function f is given by?

A
$$f(x)=2^{-1/12}\left(\dfrac{\sin\left(6x+\dfrac{\pi}{4}\right)}{\cos^2\left(6x+\dfrac{\pi}{4}\right)}\right)^{1/6}$$

B
$$f(x)=2^{-1/6}\left(\dfrac{\sin \left(6x+\dfrac{\pi}{4}\right)}{\cos^2\left(6x+\dfrac{\pi}{4}\right)}\right)^{1/12}$$

C
$$f(x)=(\sec 12x)^{1/12}(\sec 12x+\tan 12x)^{1/4}$$

D
$$f(x)=(\sec 12x)^{1/4}(\sec 12x+\tan 12x)^{1/12}$$