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CONTINUITY AND DIFFERENTIABLILITY – MCQ – LEVEL – 1

CONTINUITY AND DIFFERENTIABLILITY – MCQ – LEVEL – 1

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Question 1 of 10

1. Question
Suppose $f(x)$ is differentiable at x = 1 & $\underset{h\rightarrow 0} \lim \dfrac{1}{h}f(1+h)=5,$ then $f(1)$ equals :
- 6
- 5
- 4
- 3
Correct

Incorrect
Question 2 of 10

2. Question
If $f : (-1, 1) \rightarrow \mathbb{R}$ is a differentiable function with $f(0)=-1 ~ \& ~ f'(0)=1$ . Let $g(x) = [f(2f(x)+2)]^2,$ the $g'(0)$ is equal to :
- 0
- –2
- 4
- –4
Correct

Incorrect
Question 3 of 10

3. Question
If a function f(x) is differentiable at x = a, then $\underset{x\rightarrow 0}\lim\dfrac{x^2f(a)-a^2f(x)}{x-a}$ is:
- $2af(a)+a^2f'(a)$
- $-a^2f'(a)$
- $af(a)-a^2f'(a)$
- $2af(a)-a^2f'(a)$
Correct

Incorrect
Question 4 of 10

4. Question
If $f : D \rightarrow \mathbb{R},$ following statement is false :
- If f is differentiable, then it is continuous too
- If f is continuous, then it is differentiable too
- Polynomial function is differentiable & continuous both
- Every differentiable function is continuous
Correct

Incorrect
Question 5 of 10

5. Question
The function $y = |log x|$ is :
- Discontinuous at x = 1
- Differentiable at x = 1
- Not differentiable at x = 1
- None of these
Correct

Incorrect
Question 6 of 10

6. Question
A polynomial for all real values of x is :
- continuous
- discontinuous
- convergent
- None of these
Correct

Incorrect
Question 7 of 10

7. Question
A function $f(x) = \left \{ \begin{matrix} \dfrac{x^2 - 1}{x - 1} & ; & x \neq 1 \\ 2 & ; & x = 1 \end{matrix} \right.$ is :
- Discontinuous at x = 1
- Discontinuous at x = 2
- Continuous at x = 1
- Continuous at x = 2
Correct

Incorrect
Question 8 of 10

8. Question
Let $f(x) = \dfrac{1-\tan x}{4x-{\pi} }, ~x = \dfrac{\pi}{4}, ~x \in\left [ 0, \dfrac{\pi}{2} \right ].$ If $f(x)$ is continuous $\left [ 0, \dfrac{\pi}{2} \right ],$ in then $f\left (\dfrac{{\pi}}{4} \right )$ is :
- $1$
- $\dfrac{1}{2}$
- $-\dfrac{1}{2}$
- $-1$
Correct

Incorrect
Question 9 of 10

9. Question
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by f(x) = min {x+1, |x|+1}. Then which of the following is true?
- $f(x)\ge1 \forall x \in \mathbb{R}$
- $f(x)$ is not differentiable at x = 1
- $f(x)$ is not differentiable every where
- $f(x)$ is not differentiable at x = 0
Correct

Incorrect
Question 10 of 10

10. Question
Let $f(x)=x |x| ~ \& ~ g(x) =sin x.$

Then consider the following statements :
P : $gof$ is differentiable at x = 0.
Q : its derivative is continuous at that point.
Then which of the above statements is correct?
- P
- Q
- Both P & Q
- Neither P nor Q
Correct

Incorrect

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