JEE Advanced 2023 Paper 2 Q06 Mathematics Differentiation & Applications Differentiation Rules Hard

JEE Advanced 2023 Paper 2 · Q06 · Differentiation Rules

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

  1. A. The function $f$ is discontinuous exactly at one point in $(0, 1)$
  2. B. There is exactly one point in $(0, 1)$ at which the function $f$ is continuous but NOT differentiable
  3. C. The function $f$ is NOT differentiable at more than three points in $(0, 1)$
  4. D. The minimum value of the function $f$ is $-\dfrac{1}{512}$
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, B

Solution

Piecewise: $[4x]=0$ on $(0,1/4)$ → $f=0$; $[4x]=1$ on $[1/4,1/2)$ → $f = (x-1/4)^2(x-1/2)$; $[4x]=2$ on $[1/2,3/4)$ → $f = 2(x-1/4)^2(x-1/2)$; $[4x]=3$ on $[3/4,1)$ → $f = 3(x-1/4)^2(x-1/2)$. Continuity at $x = 1/4$: both sides give $0$ — continuous. At $x = 1/2$: left side $\to 0$ via $(1/2-1/4)^2 \cdot 0 = 0$, right side starts at $0$ — continuous. At $x = 3/4$: left side $= 2(1/2)^2(1/4) = 1/8$, right side $= 3(1/2)^2(1/4) = 3/16$ — DISCONTINUOUS. So (A) TRUE, (C) FALSE. Differentiability at $x = 1/4$: both sides smooth, derivative matches via the $(x-1/4)^2$ factor — DIFFERENTIABLE. At $x = 1/2$: $f$ continuous but the derivative changes due to $[4x]$ jump — NOT differentiable. So exactly one such point — (B) TRUE. Min by calculus on the relevant interval $f = (x-1/4)^2(x-1/2)$, minimized at $x = 5/12$ giving $-1/432$, not $-1/512$ — (D) FALSE.

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