Question

Let \( f(x) = x^2 + bx + 7 \). If \( f'(5) = 2f'\left(\frac{7}{2}\right) \), then the value of \( b \) is:

• A. \( 4 \)
• B. \( 3 \)
• C. \( -4 \)
• D. \( -3 \)
• E. \( 2 \)

Hint:

For any quadratic \( ax^2 + bx + c \), the derivative is a linear function. Linear equations are straightforward to solve once you've substituted your known values!

Answer 1

The Correct Option is C

Understanding the Concept: We first find the general derivative \( f'(x) \) of the quadratic function. Then, we substitute the specific values into the given equation to solve for the unknown coefficient \( b \).

Step 1: Find the derivative \( f'(x) \).
Given \( f(x) = x^2 + bx + 7 \). Differentiating with respect to \( x \): \[ f'(x) = 2x + b \]

Step 2: Evaluate the derivative at the given points.
For \( x = 5 \): \[ f'(5) = 2(5) + b = 10 + b \] For \( x = 7/2 \): \[ f'\left(\frac{7}{2}\right) = 2\left(\frac{7}{2}\right) + b = 7 + b \]

Step 3: Set up and solve the equation.
The condition is \( f'(5) = 2f'\left(\frac{7}{2}\right) \). \[ 10 + b = 2(7 + b) \] \[ 10 + b = 14 + 2b \] Subtract \( b \) from both sides: \[ 10 = 14 + b \] Subtract \( 14 \) from both sides: \[ b = -4 \]