Answer:
the function f(x) = -3x^2 + 12x - 9 has a maximum value of 3, which occurs at x = 2.
Explanation:
To determine whether the function f(x) = -3x^2 + 12x - 9 has a maximum or minimum, we can use the fact that a quadratic function has a maximum at the vertex if the coefficient of x^2 is negative, and a minimum at the vertex if the coefficient of x^2 is positive.
In this case, the coefficient of x^2 is -3, which is negative, so the function has a maximum at the vertex. The x-coordinate of the vertex is given by -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. Using the formula, we get:
x = -b/2a = -12/(2*(-3)) = 2
So the vertex of the parabola is at x = 2. To find the corresponding y-coordinate (i.e. the maximum value of the function), we can substitute x = 2 into the function:
f(2) = -3(2)^2 + 12(2) - 9 = -3(4) + 24 - 9 = 3
Therefore, the function f(x) = -3x^2 + 12x - 9 has a maximum value of 3, which occurs at x = 2.