Let [.] denotes the greatest integer function, f:(0,1)−{13}→R be the function defined as f(x)=[4x](x−1−[x]3)[2x](x−[2x]+14) 3+[x]. Then which of the following options is/are correct?

f(x)={0,n∈(0,14)(x−14)3,n∈[14,12)2(x−13).(x−12)3,n∈[12,34)3(x−13).(x−12)3,n∈[34,1)f(x) is discontinuous and non-differentiable at x=12,34

Let [.] denotes the greatest integer function, f:(0,1)−{13}→R be the function defined as f(x)=[4x](x−1−[x]3)[2x](x−[2x]+14) 3+[x]. Then which of the following options is/are correct?

f(x)={0,n∈(0,14)(x−14)3,n∈[14,12)2(x−13).(x−12)3,n∈[12,34)3(x−13).(x−12)3,n∈[34,1)f(x) is discontinuous and non-differentiable at x=12,34

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Let [.] denotes the greatest integer function, $f : (0, 1) - {\frac{1}{3}} \to R$ be the function defined as $f (x) = [4 x] {(x - \frac{1 - [x]}{3})}^{[2 x]} {(x - \frac{[2 x] + 1}{4})}^{^{3 + [x]}}$ . Then which of the following options is/are correct?

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The function f is discontinuous at exactly two points in its domain

There exists exactly two points 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 f is 0

answer is A, D.

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Detailed Solution

$f (x) = {\begin{cases} \begin{matrix} 0, & n \in (0, \frac{1}{4}) \\ {(x - \frac{1}{4})}^{3}, & n \in [\frac{1}{4}, \frac{1}{2}) \end{matrix} \\ \begin{matrix} 2 (x - \frac{1}{3}) . {(x - \frac{1}{2})}^{3}, & n \in [\frac{1}{2}, \frac{3}{4}) \\ 3 (x - \frac{1}{3}) . {(x - \frac{1}{2})}^{3}, & n \in [\frac{3}{4}, 1) \end{matrix} \end{cases}$