Doubt in Calculus

Answer : d
Please provide solution for this question

Just put f(x) = (any irrational no.) and put it in the sum.
The result will be of the form 51*(that irrational no.) which can't attain any of the three given values.

@Abhishek_2020_3 I have a doubt since it is said that it is a differentiable function can we even take constant value. Will it not violate that

Yaah even I though that. Because we don't know if it is a constant function

Constant function is always differentiable in R.
The function you have taken doesn't attain only irrational values.

Yeah you are correct. Got it. Thanks

But what Ayush, has taken nπ satisfies the definition of irrational numbers as

Suppose if nπ = k/l and we know k/l is rational
then it also implies that π = k/(l x n) which will be rational in that case which is definitely not possible. Thus non zero multiples of π will be irrational always.

The function he has taken will definitely be irrational for all n = Integer. But, if n = some irrational no. then we can't say if the value of function will be rational or irrational. We want a function which is irrational always.

Yaah right....gottchha