Find the minimum value of the function f(x) = 3x^2 - 12x + 7.

Find the minimum value of the function f(x) = 3x^2 - 12x + 7.

Correct Answer: 1

Step 1: Identify the function we need to analyze, which is f(x) = 3x^2 - 12x + 7.
Step 2: Recognize that this is a quadratic function in the form f(x) = ax^2 + bx + c, where a = 3, b = -12, and c = 7.
Step 3: To find the minimum value, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -(-12)/(2*3) = 12/6 = 2.
Step 5: Now that we have the x-coordinate of the vertex (x = 2), we need to find the corresponding y-coordinate by substituting x back into the function: f(2) = 3(2^2) - 12(2) + 7.
Step 6: Calculate f(2): f(2) = 3(4) - 24 + 7 = 12 - 24 + 7 = -12 + 7 = 1.
Step 7: The minimum value of the function f(x) occurs at the vertex, which is f(2) = 1.

Quadratic Functions – Understanding the properties of quadratic functions, including finding the vertex to determine minimum or maximum values.
Vertex Formula – Using the vertex formula x = -b/(2a) to find the x-coordinate of the vertex of a parabola.
Function Evaluation – Evaluating the function at the vertex to find the minimum or maximum value.