Final answer:
To find the maximum or minimum value of the function f(x) = 3(x - 3)^2 - 2, you can find the vertex of the parabola by using the formula x = -b/2a and substitute the value of x into the function to find the corresponding y-value. Alternatively, you can calculate the first derivative and set it equal to zero to find the critical points.
Step-by-step explanation:
To find the maximum or minimum value of a function, we can follow the steps below:
a) Find the vertex of the parabola to determine the maximum or minimum value. The vertex of the parabola is given by the formula x = -b/2a, where a and b are the coefficients of x in the quadratic equation. In this case, a = 3 and b = -9, so x = -(-9)/(2*3) = 3/2. To find the corresponding y-value, substitute x = 3/2 into the function: f(3/2) = 3(3/2 - 3)^2 - 2 = 3(0)^2 - 2 = -2. Therefore, the vertex is (3/2, -2), and the function has a minimum value of -2.
b) Calculate the first derivative and set it equal to zero to find the critical points. The derivative of the given function is f'(x) = 6(x - 3). Setting this equal to zero, we get 6(x - 3) = 0, which implies x - 3 = 0, and x = 3. Therefore, x = 3 is the only critical point.
Since x = 3 is also the vertex of the parabola, we can conclude that the function has a minimum value of -2 at x = 3.