Question

Let f:(0,1)→R be the function defined as f(x)=[4x](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 statements is(are) true?

• A. The function f is discontinuous exactly at the 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\(-\frac{1}{512}\)

Answer 1

The Correct Option is A, B

To analyze the function \( f(x) = [4x](x-\frac{1}{4})^2(x-\frac{1}{2}) \), where \( [x] \) denotes the greatest integer less than or equal to \( x \), follow these steps:

Discontinuity Analysis 

1. Check where \( [4x] \) may change value within \( (0,1) \). Notice \( [4x]=n \) where \( n \) is an integer, specifically \( n \in \{1, 2, 3\} \) because \( 0 < 4x < 4 \).

2. Points where changes occur are \( x = \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \). Evaluate continuity at these points:

• For \( x = \frac{1}{4}, [4x] \) jumps from 0 to 1. At \( x = \frac{1}{4} \), function switches from 0 to non-zero value. Thus, \( f \) is discontinuous.
• For \( x = \frac{1}{2}, [4x] \) jumps from 1 to 2. Similar discontinuity due to value change in \( [4x] \).
• For \( x = \frac{3}{4}, [4x] \) jumps from 2 to 3. Another discontinuity point due to the evaluation of polynomial term.

3. Therefore, \( f \) is not continuous at these three points.

Continuity and Differentiability

1. The problem asks to find exactly one point where the function is continuous but not differentiable:

• Points: \( (\frac{1}{2}, \frac{3}{4}) \) seems continuous upon inspecting polynomial terms where no jumps occur in integers.
• However, at \( x = \frac{3}{4} \), the function may smoothly transition through polynomial terms, confirming continuity without differentiability due to the modulus of polynomial not being zero.

Thus, function \( f \) is continuous at this domain but not differentiable.