Final answer:
To find the maximum value of the quadratic function f(x) = -x^2 + 3x + 2, we identify the vertex, resulting in an x-coordinate of 1.5. Substituting x = 1.5 back into the function yields the maximum value of 4.25.
Step-by-step explanation:
The function given is f(x) = -x^2 + 3x + 2, which is a quadratic equation. To find the maximum value of this function, we need to identify the vertex of the parabola. Since it is a parabola that opens downward (due to the negative coefficient before x^2), the vertex will give us the maximum value. The vertex of a parabola given by y = ax^2 + bx + c is found at the x-coordinate x = -b/(2a). For our function, a = -1 and b = 3, so the x-coordinate of the vertex is x = -3/(2*(-1)) which simplifies to x = 1.5.
Now, we substitute x = 1.5 back into the function to find the maximum value: f(1.5) = -(1.5)^2 + 3*(1.5) + 2 which simplifies to 4.25. Therefore, the maximum value of the function f(x) is 4.25, which corresponds to option a).