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Let f:(0,1)→R be the function defined as f(x)=[4x] (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?
(A)the function f is discontinuous exactly at one point in (0,1)
(B)There is exactly one point in (0,1) at which the function f is continuous but NOT differentiable
(C) The function f is NOT differentiable at more than three points in (0,1)
(D) The minimum value of the function f is -1/512
JEE Advanced 2023

This is a very conceptual JEE Advanced 2023 PYQ from Functions (Differentiability + Continuity) involving Greatest Integer Function [x].

📌 Full Question:
Let f:(0,1)\to \mathbb{R} be the function
f(x)=\left(x^2-\frac14\right)\left(x-\frac12\right)\,[4x],
\quad x\in(0,1)
where [x] is the greatest integer ≤ x.
Choose the correct statement(s):
(A) discontinuous exactly at one point
(B) exactly one point continuous but not differentiable
(C) not differentiable at more than three points
(D) minimum value is -\frac{1}{512}

In this lecture, we convert the function into a piecewise function using values of [4x] and then analyze:
• continuity points
• differentiability points
• minimum value

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