The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?

The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?

Step 1: Write down the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the first derivative f'(x) by differentiating f(x). This gives f'(x) = 3x^2 - 12x + 9.
Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 12x + 9 = 0.
Step 4: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
Step 5: Factor the quadratic equation: (x - 1)(x - 3) = 0.
Step 6: Solve for x to find critical points: x = 1 and x = 3.
Step 7: To determine if these points are local extrema, find the second derivative f''(x) = 6x - 12.
Step 8: Evaluate the second derivative at the critical points: f''(1) = 6(1) - 12 = -6 (local maximum) and f''(3) = 6(3) - 12 = 6 (local minimum).
Step 9: Conclude that there is one local maximum at x = 1 and one local minimum at x = 3.

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