Question

Which of the statement(s) is/are correct about differentiation and integration?
$\left( i \right)$ Both are operations on functions.
$\left( ii \right)$ Both satisfy the property of linearity
$\left( iii \right)$ It is not necessary that all functions are differentiable and integrable.
$\left( iv \right)$ Derivative and integral of a function when it exists is a unique function.
(a) $\left( i \right),\left( ii \right)$ and $\left( iii \right)$
(b) $\left( i \right)$ and $\left( iii \right)$
(C) Only $\left( iv \right)$
(d) All are correct

Answer 1

Hint: An operation $O$ satisfies linear property if $O\left( kf\left( x \right) \right)=kO\left( f\left( x \right) \right)$, where $k$ is a constant.

Complete step-by-step answer:
In this question, we have to find out whether the four statements are correct or incorrect. So, we will discuss these four statements one by one.
Statement $(i)$: Both are the operations on functions
This statement is true because both differentiation and integration are only applied to a function. So, they both are the operations on functions. For example, if we have a function $f\left( x \right)={{e}^{3x}}$, then,
For differentiation operator i.e. $\dfrac{d}{dx}$ ,
$\begin{aligned} & \dfrac{df\left( x \right)}{dx}=\dfrac{d{{e}^{3x}}}{dx} \\\ & \Rightarrow \dfrac{d{{e}^{3x}}}{dx}=3{{e}^{3x}} \\\ \end{aligned}$
For integration operator i.e. $\int{dx}$ ,
$\begin{aligned} & \int{f\left( x \right)dx}=\int{{{e}^{3x}}dx} \\\ & \Rightarrow \int{{{e}^{3x}}dx}=\dfrac{{{e}^{3x}}}{3} + C\\\ \end{aligned}$
Statement $\left( ii \right)$: Both satisfy the property of linearity
An operation $O$ satisfies linear property if $O\left( kf\left( x \right) \right)=kO\left( f\left( x \right) \right)$, where $k$ is a constant.
For differentiation, we have a property,
$\dfrac{d\left( kf\left( x \right) \right)}{dx}=k\dfrac{d\left( f\left( x \right) \right)}{dx}$
For integration, we have a property,
$\int{kf\left( x \right)dx=k\int{f\left( x \right)dx}}$
Hence, from the above two properties we can say that both differentiation and integration satisfy the property of linearity. So, this statement is true.
Statement $\left( iii \right)$: It is not necessary that all functions are differentiable and integrable.
It is true that not every function is differentiable or integrable.
For example,
If we consider a function $f\left( x \right)=\left| x \right|$, we can notice that this function is not differentiable at $x=0$.
If we consider a function $f\left( x \right)={{e}^{{{x}^{2}}}}$, we can notice that it is impossible for us to integrate this function i.e. this function is not integrable.
So, this statement is true.
Statement $\left( iv \right)$ Derivative and integral of a function when it exists is a unique function.
It is true that the derivative of a function is unique. But it is not true that the integral of a function is unique. The integral of a particular function can differ by a constant.
For example, if $\int{xdx}=\dfrac{{{x}^{2}}}{2}+C$ where $C$ can have any value. So, $\int{xdx}$ can be different for different values of $C$.
Hence, this statement is false.
The correct statements are $\left( i \right),\left( ii \right),\left( iii \right)$.

Note: There is a possibility that one may consider the statement $\left( iv \right)$ as a correct statement. But since the integral of a function is not unique, it is an incorrect statement.