let f x x g x x and f g x h x where 381021400

Let $$f\left( x \right) = \left[ x \right],\,g\left( x \right) = \left| x \right|$$ and $$f\left\{ {g\left( x \right)} \right\} = h\left( x \right),$$ where $$\left[ . \right]$$ is the greatest integer function. Then $$h'\left( { - 1} \right)$$ is :

A. 0

B. $$ - \infty $$

C. nonexistent

D. none of these

Answer : nonexistent

Solution :
$$\eqalign{ & h\left( x \right) = f\left\{ {g\left( x \right)} \right\} = \left[ {g\left( x \right)} \right] = \left[ {\left| x \right|} \right] \cr & h'\left( { - 1 + 0} \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {\left| { - 1 + h} \right|} \right] - \left[ {\left| { - 1} \right|} \right]}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {1 - h} \right] - \left[ 1 \right]}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{0 - 1}}{h} = - \infty \cr & h'\left( { - 1 - 0} \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {\left| { - 1 - h} \right|} \right] - \left[ {\left| { - 1} \right|} \right]}}{{ - h}} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {1 + h} \right] - \left[ 1 \right]}}{{ - h}} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{1 - 1}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} = 0 \cr} $$

Releted MCQ Question on
Calculus >> Differentiability and Differentiation

Releted Question 1

There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-

A. $$f''\left( x \right) > 0$$ for all $$x$$

B. $$ - 1 < f''\left( x \right) < 0$$ for all $$x$$

C. $$ - 2 \leqslant f''\left( x \right) \leqslant - 1$$ for all $$x$$

D. $$f''\left( x \right) < - 2$$ for all $$x$$

View Answer

Releted Question 2

If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-

A. $$-5$$

B. $$\frac{1}{5}$$

C. $$5$$

D. none of these

View Answer

Releted Question 3

Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-

A. $$8f'\left( 1 \right)$$

B. $$4f'\left( 1 \right)$$

C. $$2f'\left( 1 \right)$$

D. $$f'\left( 1 \right)$$

View Answer

Releted Question 4

Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:

A. $$\mathop {\lim }\limits_{x \to 0} f\left( x \right)$$ does not exist

B. $$f\left( x \right)$$ is continuous at $$x = 0$$

C. $$f\left( x \right)$$ is not differentiable at $$x =0$$

D. $$f'\left( 0 \right) = 1$$

View Answer

Let $$f\left( x \right) = \left[ x \right],\,g\left( x \right) = \left| x \right|$$ and $$f\left\{ {g\left( x \right)} \right\} = h\left( x \right),$$ where $$\left[ . \right]$$ is the greatest integer function. Then $$h'\left( { - 1} \right)$$ is :

Releted MCQ Question on
Calculus >> Differentiability and Differentiation

There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-

If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-

Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-

Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:

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Let $$f\left( x \right) = \left[ x \right],\,g\left( x \right) = \left| x \right|$$ and $$f\left\{ {g\left( x \right)} \right\} = h\left( x \right),$$ where $$\left[ . \right]$$ is the greatest integer function. Then $$h'\left( { - 1} \right)$$ is :

Releted MCQ Question on Calculus >> Differentiability and Differentiation

There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-

If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-

Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-

Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:

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