Final answer:
The function f(x) = -(1)/(2)(x-4)^2 + 1 has a maximum at the point (4, 1). This is because it is a downward opening parabola indicated by the negative coefficient in front of the squared term.
Step-by-step explanation:
The student has asked whether the function f(x) = -(1)/(2)(x-4)^2 + 1 has a maximum or minimum. To determine this, we must examine the form of the function and employ calculus, specifically differentiating the function to find its critical points and determine their nature. Given the negative coefficient in front of the squared term, this implies that the function opens downwards and hence has a maximum point.
Let's find the vertex of the parabola. Since the function is in the form -A(x-h)^2 + k, the vertex is located at (h, k), which in this case is (4, 1). This point is actually the maximum of the function because the function opens downwards due to the negative coefficient in front of the quadratic term. To confirm, we can derive the function to get f'(x), and find that for x = 4, the derivative is zero, which is a critical point. Since the coefficient of the squared term is negative, this critical point is a maximum.
Therefore, the function f(x) has a maximum value at x=4, and this maximum value is f(4) = 1.