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

The function is defined as:

\( f(x) = -2x^2 + 4x + 1 \)

Step 1: Determine the critical point via the first derivative

To locate the maximum or minimum, we compute the derivative of \( f(x) \): \[ f'(x) = \frac{d}{dx}(-2x^2 + 4x + 1) \] \[ f'(x) = -4x + 4 \] Equating the derivative to zero yields the critical point: \[ -4x + 4 = 0 \] \[ -4x = -4 \] \[ x = 1 \]

Step 2: Confirm the nature of the critical point

Given that the coefficient of \( x^2 \) in the quadratic function is negative (-2), the parabola's orientation is downwards, indicating that the critical point corresponds to a maximum.

✅ Final Answer:

The function attains its maximum value at \( \boxed{x = 1} \).