Let `f(x)=|x| and g(x)=[x]`, (where `[.]` denotes the greatest integer function) Then, `(fog)'(-1)` is

Options

A. 0
B. does not exist
C. `-1`
D. 1

Answer

To find \((f \circ g)'(-1)\), we will first determine the functions \(f(x)\) and \(g(x)\) and then find the composition of these functions.

1. **Define the Functions:**
- \(f(x) = |x|\) (the absolute value function)
- \(g(x) = [x]\) (the greatest integer function)

2. **Find the Composition \(f(g(x))\):**
- The composition \(f(g(x))\) means we substitute \(g(x)\) into \(f(x)\):
\[ f(g(x)) = f([x]) = |[x]| \]
This means we take the greatest integer less than or equal to \(x\) and then take the absolute value of that integer.

3. **Evaluate \(g(-1)\):**
- We find \(g(-1)\):
\[ g(-1) = [-1] = -1 \]

4. **Evaluate \(f(g(-1))\):**
- Now we substitute \(g(-1)\) into \(f\):
\[ f(g(-1)) = f(-1) = |-1| = 1 \]

5. **Find the Derivative \((f \circ g)'(x)\):**
- To find the derivative of \(f(g(x)) = |[x]|\), we need to analyze the function \(g(x)\) and its behavior around \(x = -1\).
- The greatest integer function \([x]\) is not continuous at integer values, and specifically, it has a jump discontinuity at \(x = -1\).

6. **Check the Continuity at \(x = -1\):**
- The function \(f(g(x)) = |[x]|\) is not continuous at \(x = -1\) because:
- As \(x\) approaches \(-1\) from the left, \([x]\) approaches \(-2\) (thus \(|[x]| \to 2\)).
- As \(x\) approaches \(-1\) from the right, \([x]\) approaches \(-1\) (thus \(|[x]| \to 1\)).
- Since the left-hand limit and the right-hand limit do not match, \(|[x]|\) is discontinuous at \(x = -1\).

7. **Conclusion About the Derivative:**
- Since the function \(|[x]|\) is discontinuous at \(x = -1\), the derivative \((f \circ g)'(-1)\) does not exist.

Thus, the final answer is:
\[ (f \circ g)'(-1) \text{ does not exist.} \]