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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.
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Practice Questions
1 question
Q1
Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.
(2, 3)
(3, 0)
(1, 1)
(0, 9)
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f'(x) = 6x - 12. Setting f'(x) = 0 gives x = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.
Solution:
f'(x) = 6x - 12. Setting f'(x) = 0 gives x = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Steps: 8
Show Steps
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).
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