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 \((f \circ g)'(-1)\) where \(f(x) = |x|\) and \(g(x) = [x]\) (the greatest integer function), we will follow these steps:

### Step 1: Define the composition of functions
We start by defining the composition of the functions \(f\) and \(g\):
\[ f(g(x)) = f([x]) = |[x]| \]
This means that we are taking the greatest integer of \(x\) and then applying the absolute value function to it.

### Step 2: Evaluate \(g(-1)\)
Next, we need to evaluate \(g(-1)\):
\[ g(-1) = [-1] = -1 \]
The greatest integer function at \(-1\) returns \(-1\).

### Step 3: Evaluate \(f(g(-1))\)
Now we evaluate \(f(g(-1))\):
\[ f(g(-1)) = f(-1) = |-1| = 1 \]

### Step 4: Find the derivative of \(f(g(x))\)
To find \((f \circ g)'(-1)\), we need to differentiate \(f(g(x)) = |[x]|\). The derivative of \(|x|\) is:
\[ \frac{d}{dx} |x| = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \\ \text{undefined} & \text{if } x = 0 \end{cases} \]
Since \([x]\) is a piecewise constant function, the derivative \(g'(x)\) is \(0\) except at integer points where it is undefined.

### Step 5: Evaluate the derivative at \(x = -1\)
At \(x = -1\), \(g(x)\) is constant in the interval \([-1, 0)\) and equals \(-1\). Therefore, we need to evaluate:
\[ (f \circ g)'(-1) = f'(g(-1)) \cdot g'(-1) \]
We already found:
- \(g(-1) = -1\)
- \(f'(-1) = -1\) (since \(-1 < 0\))
- \(g'(-1)\) is undefined.

Thus, we have:
\[ (f \circ g)'(-1) = f'(-1) \cdot g'(-1) = -1 \cdot \text{undefined} \]
This means that the derivative does not exist at \(x = -1\).

### Conclusion
Therefore, \((f \circ g)'(-1)\) does not exist.

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