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If f(x)={{:(x + 1, x gt 1),(0",", x=1 "t...

If `f(x)={{:(x + 1, x gt 1),(0",", x=1 "then f'(0) equal to "),(7-3x"," , x lt 1):}`

A

`-1`

B

`-2`

C

`-3`

D

`-4`

Text Solution

AI Generated Solution

The correct Answer is:
To find \( f'(0) \) for the given piecewise function \( f(x) \), we need to analyze the function at \( x = 0 \). ### Step-by-Step Solution: 1. **Identify the Piecewise Function**: The function is defined as: \[ f(x) = \begin{cases} x + 1 & \text{if } x > 1 \\ 0 & \text{if } x = 1 \\ 7 - 3x & \text{if } x < 1 \end{cases} \] 2. **Determine the Relevant Piece for \( x = 0 \)**: Since \( 0 < 1 \), we use the piece \( f(x) = 7 - 3x \). 3. **Differentiate the Relevant Piece**: We need to find the derivative of \( f(x) = 7 - 3x \). \[ f'(x) = \frac{d}{dx}(7 - 3x) = 0 - 3 = -3 \] 4. **Evaluate the Derivative at \( x = 0 \)**: Since \( f'(x) = -3 \) is constant (independent of \( x \)), we have: \[ f'(0) = -3 \] ### Final Answer: Thus, \( f'(0) = -3 \). ---
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