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if f(x)=2x^(2)+3x-5, the what is f'(0)+3...

if `f(x)=2x^(2)+3x-5`, the what is `f'(0)+3f'(-1)=`

A

`-1`

B

0

C

1

D

2

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find \( f'(0) + 3f'(-1) \) where \( f(x) = 2x^2 + 3x - 5 \). ### Step 1: Differentiate \( f(x) \) First, we differentiate the function \( f(x) \) with respect to \( x \). \[ f'(x) = \frac{d}{dx}(2x^2 + 3x - 5) \] Using the power rule, we differentiate each term: - The derivative of \( 2x^2 \) is \( 2 \cdot 2x^{2-1} = 4x \). - The derivative of \( 3x \) is \( 3 \). - The derivative of a constant \( -5 \) is \( 0 \). Thus, we have: \[ f'(x) = 4x + 3 \] ### Step 2: Calculate \( f'(0) \) Now we substitute \( x = 0 \) into \( f'(x) \): \[ f'(0) = 4(0) + 3 = 3 \] ### Step 3: Calculate \( f'(-1) \) Next, we substitute \( x = -1 \) into \( f'(x) \): \[ f'(-1) = 4(-1) + 3 = -4 + 3 = -1 \] ### Step 4: Calculate \( f'(0) + 3f'(-1) \) Now we can compute \( f'(0) + 3f'(-1) \): \[ f'(0) + 3f'(-1) = 3 + 3(-1) = 3 - 3 = 0 \] ### Final Answer Thus, the final result is: \[ f'(0) + 3f'(-1) = 0 \]
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