The greatest integer function defined by \( f(x) = [x] \), \( 1 < x < 3 \) is not differentiable at \( x = \):
- (A) 0
- (B) 1
- (C) 2
- (D) 3/2
The Correct Option is C
Solution and Explanation
Step 1: Understand the greatest integer function
The greatest integer function, denoted by [x], returns the greatest integer less than or equal to x. It is also called the floor function.
Step 2: Behavior of the greatest integer function
The function is constant within intervals between integers but jumps abruptly at integer points. For example, [1.5] = 1, [2.3] = 2, but at x = 2, the function jumps from 1 to 2.
Step 3: Differentiability and continuity
Differentiability requires the function to be continuous and smooth at the point. However, the greatest integer function has jump discontinuities at all integer points, making it non-differentiable there.
Step 4: Analyze the interval 1 < x < 3
Within the interval, the function jumps at integers 2 and 3. Hence, the points where the function is not differentiable are x = 2 and x = 3.
Step 5: Conclusion
Since the question asks where the function is not differentiable within 1 < x < 3, the answer is x = 2 (as 3 is excluded if we consider open interval).
Final Answer: (C) 2
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