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Let, $f(x)=x-[x];x\in R$, where [.] represents the greatest integer function, then $f'\left( \dfrac{1}{2} \right)$ is equal to,
a) 1
b) 0
c) -1
d) 2

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Last updated date: 28th Feb 2024
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IVSAT 2024
Answer
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Hint: Use the formula given below to solve the problem,
$[x]=x$ If ‘x’ is an integer
$\dfrac{d}{dx}[x]=0$ If x is a fraction
$\dfrac{d}{dx}[x]=\infty $ If x is an integer

Complete step by step answer:
We will write given equation first,
$f(x)=x-[x]$
As [.] is a greatest integer function therefore we should know the formula to find derivative of greatest integer function before finding the derivative of f(x),
Formula:
$\dfrac{d}{dx}[x]=0$ If x is a fraction…………………………….. (1)
$\dfrac{d}{dx}[x]=\infty $ If x is an integer
Now,
$f(x)=x-[x]$
Differentiating f(x) with respect to x,
$\therefore f'(x)=\dfrac{d}{dx}\left( x-[x] \right)$
If we differentiate the two terms separately we will get,
\[\therefore f'(x)=\dfrac{d}{dx}x-\dfrac{d}{dx}[x]\]
As we know the formula to find the derivative of ‘x’ which is \[\dfrac{d}{dx}x=1\],
\[\therefore f'(x)=1-\dfrac{d}{dx}[x]\]
Here, we have to find the derivative of $f(x)$ at $x=\dfrac{1}{2}$
As $x=\dfrac{1}{2}$ is a fraction therefore we can use the formula number 1 so that we can directly write, \[f'(\dfrac{1}{2})\]
\[\therefore f'(\dfrac{1}{2})=1-0\]
\[\therefore f'(\dfrac{1}{2})=1\]
Therefore, we will get the final answer as \[f'(\dfrac{1}{2})\] is 1
Hence, the correct answer is option (a).

Note:
While calculating derivatives of greatest integer function always check whether the value of x is fraction or an integer.
If we don’t know the formula of derivative of greatest integer function then we should at least know the definition which is given below to solve this type of problems,
If f(x) = [x] then,
$[x]=0~$ If ‘x’ is a fraction
$[x]=x~$ If ‘x’ is an integer