Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a

Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.

Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.

Correct Answer: (2, 3)

Step 1: Write down the function f(x) = 3x^2 - 12x + 9.
Step 2: Find the derivative of the function, which is f'(x). The derivative tells us the slope of the function.
Step 3: Calculate the derivative: f'(x) = 6x - 12.
Step 4: Set the derivative equal to zero to find critical points: 6x - 12 = 0.
Step 5: Solve for x: Add 12 to both sides to get 6x = 12, then divide by 6 to find x = 2.
Step 6: Now, we need to find the value of the function at x = 2. Substitute x = 2 into the original function: f(2) = 3(2^2) - 12(2) + 9.
Step 7: Calculate f(2): f(2) = 3(4) - 24 + 9 = 12 - 24 + 9 = -3.
Step 8: The coordinates of the point where the function has a local maximum are (2, -3).

Finding Local Extrema – The question tests the ability to find local maxima or minima of a quadratic function using calculus, specifically by finding the derivative and setting it to zero.
Evaluating Functions – It also tests the skill of evaluating the function at critical points to determine the function's value at those points.