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Determine the critical points of the function f(x) = x^3 - 6x^2 + 9x.
Determine the critical points of the function f(x) = x^3 - 6x^2 + 9x.
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Practice Questions
1 question
Q1
Determine the critical points of the function f(x) = x^3 - 6x^2 + 9x.
(0, 0)
(1, 4)
(2, 0)
(3, 0)
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f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 1)(x - 3) = 0, so critical points are x = 1 and x = 3.
Questions & Step-by-step Solutions
1 item
Q
Q: Determine the critical points of the function f(x) = x^3 - 6x^2 + 9x.
Solution:
f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 1)(x - 3) = 0, so critical points are x = 1 and x = 3.
Steps: 7
Show Steps
Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
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) = 3x^2 - 12x + 9.
Step 4: Set the derivative equal to zero to find critical points: 3x^2 - 12x + 9 = 0.
Step 5: Factor the equation: (x - 1)(x - 3) = 0.
Step 6: Solve for x by setting each factor equal to zero: x - 1 = 0 gives x = 1, and x - 3 = 0 gives x = 3.
Step 7: The critical points are x = 1 and x = 3.
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