Question Detail
Question
Find the maximum value of the function \(y = \frac{x^2}{2} - 2x + 2\).
Correct Answer
Option D is the correct answer.
Detailed Explanation
This is a quadratic function opening upwards (coefficient of x² is positive 1/2), so it has a minimum, not a maximum. There might be a domain restriction not stated, or the question intends to find the minimum. For a quadratic y = ax² + bx + c, the vertex occurs at x = -b/(2a). Here, a=1/2, b=-2, so x = -(-2)/(2*(1/2)) = 2/1 = 2. The value at x=2 is y = (4/2) - 4 + 2 = 2 - 4 + 2 = 0. If the domain is all real numbers, the minimum is 0 and there is no maximum. Among the options, 0 is the minimum value. Option D is selected as the most plausible answer given the options.
Hint
The vertex of a parabola y = ax² + bx + c gives its minimum (if a>0) or maximum (if a<0). The x-coordinate of the vertex is -b/(2a).