If for `f(x) = kx^(2) + 5x + 3, f'(2) = 6`, then k is equal to

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(3)`
D. 1

Answer

To solve the problem, we need to find the value of \( k \) given that \( f(x) = kx^2 + 5x + 3 \) and \( f'(2) = 6 \).

### Step-by-Step Solution:

1. **Write down the function**:
\[ f(x) = kx^2 + 5x + 3 \]

2. **Differentiate the function**:
To find \( f'(x) \), we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(kx^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(3) \]
Using the power rule:
\[ f'(x) = 2kx + 5 + 0 = 2kx + 5 \]

3. **Substitute \( x = 2 \) into the derivative**:
We need to find \( f'(2) \):
\[ f'(2) = 2k(2) + 5 \]
Simplifying this gives:
\[ f'(2) = 4k + 5 \]

4. **Set the derivative equal to 6**:
According to the problem, \( f'(2) = 6 \):
\[ 4k + 5 = 6 \]

5. **Solve for \( k \)**:
Subtract 5 from both sides:
\[ 4k = 6 - 5 \]
\[ 4k = 1 \]
Now, divide both sides by 4:
\[ k = \frac{1}{4} \]

### Final Answer:
Thus, the value of \( k \) is:
\[ k = \frac{1}{4} \]