What is the minimum value of the function f(x) = 3x^2 - 12x + 7? (2019)

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Question: What is the minimum value of the function f(x) = 3x^2 - 12x + 7? (2019)

Options:

Correct Answer: -1

Exam Year: 2019

Solution:

The minimum occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 7 = 1.

What is the minimum value of the function f(x) = 3x^2 - 12x + 7? (2019)

What is the minimum value of the function f(x) = 3x^2 - 12x + 7? (2019)

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 of ax^2 + bx + c, where a = 3, b = -12, and c = 7.
Step 3: To find the minimum value of a quadratic function, use the formula for the x-coordinate of the vertex: x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -(-12)/(2*3).
Step 5: Simplify the expression: x = 12/(6) = 2.
Step 6: Now, substitute x = 2 back into the original function to find the minimum value: f(2) = 3(2^2) - 12(2) + 7.
Step 7: Calculate f(2): f(2) = 3(4) - 24 + 7 = 12 - 24 + 7 = -12 + 7 = -5.
Step 8: Therefore, the minimum value of the function f(x) is -5.

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