Question:
\([\,]\) represents the greatest integer function. At \(x=-1\),
\[
\frac{d}{dx}\sin \pi [x]
\]
is
\([\,]\) represents the greatest integer function. At \(x=-1\),
\[
\frac{d}{dx}\sin \pi [x]
\]
is
Show Hint
For greatest integer functions, first determine the interval in which the function remains constant. A constant function always has derivative zero on that interval.
Updated On: Jun 26, 2026
- \(0\)
- \(2\)
- \(-2\)
- \(\frac{1}{2}\)
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The Correct Option is A
Solution and Explanation
Step 1: Define the given function.
The given function is \[ f(x)=\sin \pi [x] \] where \[ [x] \] denotes the greatest integer function.
Step 2: Evaluate the function near \(x=-1\).
For \[ -1\leq x\lt 0, \] we have \[ [x]=-1 \] Thus, \[ f(x)=\sin(-\pi) \] Since \[ \sin(-\pi)=0, \] we get \[ f(x)=0 \] For \[ -2\lt x\lt -1, \] we have \[ [x]=-2 \] Thus, \[ f(x)=\sin(-2\pi)=0 \] Hence, around \(x=-1\), the function remains constant and equal to zero.
Step 3: Find the derivative at \(x=-1\).
Since the function is constant in a neighborhood around \(x=-1\), \[ f'(x)=0 \] Therefore, \[ f'(-1)=0 \]
Step 4: Final conclusion.
Hence, \[ \boxed{0} \]
The given function is \[ f(x)=\sin \pi [x] \] where \[ [x] \] denotes the greatest integer function.
Step 2: Evaluate the function near \(x=-1\).
For \[ -1\leq x\lt 0, \] we have \[ [x]=-1 \] Thus, \[ f(x)=\sin(-\pi) \] Since \[ \sin(-\pi)=0, \] we get \[ f(x)=0 \] For \[ -2\lt x\lt -1, \] we have \[ [x]=-2 \] Thus, \[ f(x)=\sin(-2\pi)=0 \] Hence, around \(x=-1\), the function remains constant and equal to zero.
Step 3: Find the derivative at \(x=-1\).
Since the function is constant in a neighborhood around \(x=-1\), \[ f'(x)=0 \] Therefore, \[ f'(-1)=0 \]
Step 4: Final conclusion.
Hence, \[ \boxed{0} \]
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