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Find the critical points of f(x) = x^3 - 3x^2 + 4.
Find the critical points of f(x) = x^3 - 3x^2 + 4.
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Practice Questions
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Q1
Find the critical points of f(x) = x^3 - 3x^2 + 4.
(0, 4)
(1, 2)
(2, 0)
(3, 1)
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Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2. Evaluating f(1) = 2.
Questions & Step-by-step Solutions
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Q
Q: Find the critical points of f(x) = x^3 - 3x^2 + 4.
Solution:
Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2. Evaluating f(1) = 2.
Steps: 8
Show Steps
Step 1: Start with the function f(x) = x^3 - 3x^2 + 4.
Step 2: Find the derivative of the function, which is f'(x). The derivative of f(x) is f'(x) = 3x^2 - 6x.
Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 6x = 0.
Step 4: Factor the equation: 3x(x - 2) = 0.
Step 5: Solve for x by setting each factor equal to zero: 3x = 0 gives x = 0, and x - 2 = 0 gives x = 2.
Step 6: The critical points are x = 0 and x = 2.
Step 7: To find the value of the function at a point, choose a point between the critical points, like x = 1.
Step 8: Evaluate f(1) = 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.
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