Let f:(0,1) rightarrow mathbb R be the function defined as f(x)=[4 x] (x-1/4 ) 2 (x-1/2 ) , where [x] denotes the greatest integer less than or equal to x . Then which of the following statements is(are) true?

The function f is discontinuous exactly at one point in (0,1)

There is exactly one point in (0,1) at which the function f is continuous but NOT differentiable

The function f is NOT differentiable at more than three points in (0,1)

The minimum value of the function f is -1/512

Mathematics · Limits, Continuity and Differentiability

JEE Advanced 2023 — Paper 2 — Question 6

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^{2}\left(x-\frac{1}{2}\right)$ , where $[x]$ denotes the greatest

integer less than or equal to x . Then which of the following statements is(are) true?

Option A:
The function $f$ is discontinuous exactly at one point in $(0,1)$
Correct
Option B:
There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable
Correct
Option C:
The function f is NOT differentiable at more than three points in $(0,1)$
Option D:
The minimum value of the function f is $-\frac{1}{512}$

Answer: A, B

Step-by-step solution

The function is $f(x) = [4x] (x-\frac14)^2 (x-\frac12)$ on $(0,1)$ . Break into intervals: $[4x]=0$ for $0\frac12$ , $f(x)>0$ .

So minimum is $-\frac1{432}$ , not $-\frac1{512}$ , so statement D is false. Thus only A and B are true.

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2023
Paper: Paper 2
Subject: Mathematics
Chapter: Limits, Continuity and Differentiability
Topic: Continuity