What is the value of f '' ( x ): if f ( x ) = `3 x^ 2 - 7 x - 5`

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the value of \( f''(x) \) for the function \( f(x) = 3x^2 - 7x - 5 \), we will differentiate the function twice. Here’s the step-by-step solution:

### Step 1: Differentiate \( f(x) \) to find \( f'(x) \)

The function is given as:
\[ f(x) = 3x^2 - 7x - 5 \]

To find the first derivative \( f'(x) \), we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(7x) - \frac{d}{dx}(5) \]

Calculating each term:
- The derivative of \( 3x^2 \) is \( 6x \) (using the power rule).
- The derivative of \( -7x \) is \( -7 \).
- The derivative of \( -5 \) (a constant) is \( 0 \).

Thus, we have:
\[ f'(x) = 6x - 7 \]

### Step 2: Differentiate \( f'(x) \) to find \( f''(x) \)

Now, we differentiate \( f'(x) \) to find the second derivative \( f''(x) \):
\[ f''(x) = \frac{d}{dx}(6x - 7) \]

Calculating each term:
- The derivative of \( 6x \) is \( 6 \).
- The derivative of \( -7 \) (a constant) is \( 0 \).

Thus, we have:
\[ f''(x) = 6 \]

### Final Answer

The value of \( f''(x) \) is:
\[ \boxed{6} \]

---