Final answer:
The maximum value of the function f(x) = -3x^2 + 18x + 10 is at the point (3, 37) because this is a downward-opening parabola, and the vertex of the parabola represents the maximum value of the function.
Step-by-step explanation:
To identify the maximum or minimum value of the function f(x) = -3x^2 + 18x + 10, we can first recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -3, b = 18, and c = 10. The coefficient of x^2 (a) is negative, indicating that the parabola opens downwards, hence the function has a maximum value rather than a minimum.
To find the x-coordinate of the vertex, we use the formula -b/(2a), which gives us -18/(2×(-3)) = 3. Substituting x = 3 back into the function, we get f(3) = -3(3)^2 + 18(3) + 10 = -27 + 54 + 10 = 37. Hence, the maximum value of the function is at the point (3, 37).