Question

The maximum value of the function \( f(x) = -2x^2 + 4x + 1 \) occurs at:

• A. \( x = 1 \)
• B. \( x = -1 \)
• C. \( x = 0 \)
• D. \( x = 2 \)

Hint:

To find the maximum or minimum of a function, first find the critical points by setting the first derivative equal to zero. Use the second derivative to determine whether it's a maximum or minimum.

Answer 1

The Correct Option is A

We are given the function \( f(x) = -2x^2 + 4x + 1 \), and we need to find the value of \( x \) at which the function reaches its maximum. Step 1: Find the first derivative of the function To find the critical points, we first need to differentiate the function: \[ f'(x) = \frac{d}{dx} \left( -2x^2 + 4x + 1 \right) \] Using standard differentiation rules, we get: \[ f'(x) = -4x + 4 \] Step 2: Set the first derivative equal to zero to find the critical points Set \( f'(x) = 0 \) to find the critical points: \[ -4x + 4 = 0 \] Solving for \( x \), we get: \[ x = 1 \] Step 3: Verify whether this is a maximum or minimum To confirm that \( x = 1 \) corresponds to a maximum, we check the second derivative: \[ f''(x) = \frac{d}{dx} (-4x + 4) = -4 \] Since \( f''(x) = -4 \) is negative, the function is concave down, indicating that \( x = 1 \) corresponds to a maximum. Answer: The maximum value of the function occurs at \( x = 1 \), so the correct answer is option (1).