Let f(x) be a function defined in `1lexltoo` by.
`f(x)={:[(2-x,"for "1lexle2),(3x-x^(2),"for "xgt2):}`
What is the differentiable coefficient of f(x) at x = 3?

Options

A. A) 1
B. B) 2
C. C) `-1`
D. D) `-3`

Answer

To find the differentiable coefficient (derivative) of the function \( f(x) \) at \( x = 3 \), we will follow these steps:

### Step 1: Identify the function for \( x = 3 \)
Since \( x = 3 \) is greater than 2, we will use the second part of the piecewise function:
\[ f(x) = 3x - x^2 \quad \text{for } x > 2 \]

### Step 2: Differentiate the function
Now, we need to find the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(3x - x^2) \]
Using the power rule:
\[ f'(x) = 3 - 2x \]

### Step 3: Evaluate the derivative at \( x = 3 \)
Now, we substitute \( x = 3 \) into the derivative to find the differentiable coefficient:
\[ f'(3) = 3 - 2(3) \]
Calculating this gives:
\[ f'(3) = 3 - 6 = -3 \]

### Conclusion
The differentiable coefficient of \( f(x) \) at \( x = 3 \) is:
\[ \boxed{-3} \]

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