Final answer:
The quadratic function f(x) = 3x² - 12x + 1 has a minimum value at the point (2, -11), found by determining the vertex of the parabola.
Step-by-step explanation:
To find the maximum or minimum of the quadratic function f(x) = 3x² - 12x + 1, we need to determine the vertex of the parabola represented by the function. Since the coefficient of x² is positive, the parabola opens upwards, indicating that the vertex is a minimum point.
The vertex of a quadratic function in the form ax² + bx + c can be found using the formula -b/(2a) for the x-coordinate. Plugging in our coefficients gives:
x = -(-12) / (2 * 3) = 12/6 = 2
To find the y-coordinate of the vertex, substitute the x-value back into the function:
f(2) = 3(2)² - 12(2) + 1 = 12 - 24 + 1 = -11
Thus, the vertex of the parabola, and the minimum point of the function, is at (2, -11). So, the function has a minimum value of -11 at x = 2.