Question

Share

If a real valued function $f(x)$ is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
For example, every polynomial, constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
Based on the above information, answer the following questions.

1. If $\text{f}(\text{x})=\begin{cases}\text{x},\text{for x}\leq0\\0,\text{for x}>0\end{cases},$ then at $x = 0$

1. $f(x)$ is differentiable and continuous.
2. $f(x)$ is neither continuous nor differentiable.
3. $f(x)$ is continuous but not differentiable.
4. None of these.

2. If $\text{f}(\text{x})=|\text{x}-1|,\text{x }\in\text{ R},$ then at $x = 1$

1. $f(x)$ is not continuous.
2. $f(x)$ is continuous but not differentiable.
3. $f(x)$ is continuous and differentiable.
4. None of these.

3. $f(x) = x^3$ is:

1. Continuous but not differentiable at $x = 3$
2. Continuous but not differentiable at $x = 3$
3. Neither continuous nor differentiable at $x = 3$
4. None of these.

4. If $\text{f}(\text{x})=[\sin\text{x}],$ then which of the following is true?

1. $f(x)$ is continuous and differentiable at $x = 0.$
2. $f(x)$ is discontinuous at $x = 0.$
3. $f(x)$ is continuous at $x = 0$ but not differentiable.
4. $f(x)$ is differentiable but not continuous at $\text{x}=\frac{\pi}{2}.$

5. If $\text{f}(\text{x})=\sin^{-1}\text{x},-1\leq\text{x}\leq1,$ then:

1. $f(x)$ is both continuous and differentiable.
2. $f(x)$ is neither continuous nor differentiable.
3. $f(x)$ is continuous but not differentiable.
4. None of these.