Let f:(0,1)tomathbbR be the function defined as f(x)=[4x] (x-frac14 )^2 (x-frac12 ) , where [x] denotes the greatest integer less than or equal to x . Then which of the following statements is(are) true?

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

Limits & Continuity: Let f:(0,1)tomathbbR be the function defined as f(x)=[4x] (x-frac14… | JEE Advanced 2023 Paper 2

JEE Advanced 2023 Paper 2Question 1

Let $f:(0,1)\to\mathbb{R}$ be the function defined as $f(x)=[4x]\left(x-\frac14\right)^2\left(x-\frac12\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?

Limits & Continuity

More than one option may be correct.

The function $f$ is discontinuous exactly at one point in $(0,1)$✓ Correct answerThere is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable✓ Correct answerThe function $f$ is NOT differentiable at more than three points in $(0,1)$The minimum value of the function $f$ is $-\frac{1}{512}$

Solution

$[4x]$ jumps at $\frac14,\frac12,\frac34$; at $\frac14$ the factor $(x-\frac14)^2$ makes $f$ differentiable, at $\frac12$ $f$ is continuous but one-sided derivatives are $\frac1{16}$ and $\frac2{16}$, at $\frac34$ $f$ is discontinuous. On $[\frac14,\frac12)$ the minimum is at $x=\frac5{12}$: $-\frac1{432}$. So A and B are true.

Limits & Continuity - JEE Mathematics question (JEE Advanced 2023 Paper 2)

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