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

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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 $- \frac{1}{512}$

answer is A, B.

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Detailed Solution

$f (x) = \{\begin{matrix} 0 & 0 < x < \frac{1}{4} \\ {(X - \frac{1}{4})}^{2} (X - \frac{1}{2}) & \frac{1}{4} \leq x < \frac{1}{2} \\ 2 {(X - \frac{1}{4})}^{2} (X - \frac{1}{2}) & \frac{1}{2} \leq x < \frac{3}{4} \\ 3 {(X - \frac{1}{4})}^{2} (X - \frac{1}{2}) & \frac{3}{4} \leq x < 1 \end{matrix}$
Continuoual at $x = \frac{1}{4}, \frac{1}{2}$
diss $at x = \frac{3}{4}$
not diffat $x = \frac{1}{2}$
min of $f (x) = \frac{- 1}{432} at x = \frac{5}{12}$