The left-hand derivative of `f(x) =[x]sin (pix)` at k an interger, is:

Options

A. `(-1)^(k)(k-1) pi`
B. `(-1)^(k-1)(k-1)pi`
C. `(-1)^(k) kpi`
D. `(-1)^(k-1)kpi`

Answer

To find the left-hand derivative of the function \( f(x) = [x] \sin(\pi x) \) at \( k \), where \( k \) is an integer, we will follow these steps:

### Step 1: Understand the Left-Hand Derivative
The left-hand derivative of a function \( f \) at a point \( k \) is defined as:
\[ f'_-(k) = \lim_{h \to 0^-} \frac{f(k + h) - f(k)}{h} \]
This means we will consider values of \( h \) approaching 0 from the left (negative side).

### Step 2: Substitute in the Definition
We substitute \( f(k + h) \) and \( f(k) \):
\[ f(k + h) = [k + h] \sin(\pi(k + h)) \]
Since \( k \) is an integer and \( h \) is approaching 0 from the left, \( [k + h] = k \) when \( h \) is very small and negative. Thus:
\[ f(k + h) = k \sin(\pi(k + h)) \]
And:
\[ f(k) = [k] \sin(\pi k) = k \sin(\pi k) = k \cdot 0 = 0 \quad \text{(since } \sin(\pi k) = 0 \text{ for any integer } k\text{)} \]

### Step 3: Write the Limit Expression
Now we can write the limit expression for the left-hand derivative:
\[ f'_-(k) = \lim_{h \to 0^-} \frac{k \sin(\pi(k + h)) - 0}{h} = \lim_{h \to 0^-} \frac{k \sin(\pi(k + h))}{h} \]

### Step 4: Simplify the Sine Function
Using the sine addition formula:
\[ \sin(\pi(k + h)) = \sin(\pi k + \pi h) = \sin(\pi k) \cos(\pi h) + \cos(\pi k) \sin(\pi h) \]
Since \( \sin(\pi k) = 0 \), we have:
\[ \sin(\pi(k + h)) = \cos(\pi k) \sin(\pi h) \]
For integer \( k \), \( \cos(\pi k) = (-1)^k \). Therefore:
\[ \sin(\pi(k + h)) = (-1)^k \sin(\pi h) \]

### Step 5: Substitute Back into the Limit
Now substituting back into the limit:
\[ f'_-(k) = \lim_{h \to 0^-} \frac{k (-1)^k \sin(\pi h)}{h} \]

### Step 6: Apply the Limit
Using the limit property \( \lim_{h \to 0} \frac{\sin(\pi h)}{\pi h} = 1 \):
\[ f'_-(k) = k (-1)^k \cdot \pi \cdot \lim_{h \to 0^-} \frac{\sin(\pi h)}{\pi h} = k (-1)^k \]

### Final Answer
Thus, the left-hand derivative of \( f(x) = [x] \sin(\pi x) \) at \( k \) is:
\[ f'_-(k) = k (-1)^k \]

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