For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function i

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Question: For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function is increasing.

Options:

Correct Answer: (0, 3)

Solution:

f\'(x) = 3x^2 - 12x + 9. The critical points are x = 1 and x = 3. The function is increasing on (1, 3) and (3, ∞).

For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function is increasing.

For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function is increasing.

Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the derivative of the function, f'(x), to determine where the function is increasing or decreasing.
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: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
Step 6: Factor the quadratic equation: (x - 1)(x - 3) = 0.
Step 7: Solve for x to find the critical points: x = 1 and x = 3.
Step 8: Use the critical points to test intervals: (-∞, 1), (1, 3), and (3, ∞).
Step 9: Choose a test point in each interval and plug it into f'(x) to see if the derivative is positive (increasing) or negative (decreasing).
Step 10: For the interval (-∞, 1), choose x = 0: f'(0) = 9 (positive, so increasing).
Step 11: For the interval (1, 3), choose x = 2: f'(2) = 3(2^2) - 12(2) + 9 = 3 (positive, so increasing).
Step 12: For the interval (3, ∞), choose x = 4: f'(4) = 3(4^2) - 12(4) + 9 = 9 (positive, so increasing).
Step 13: Conclude that the function is increasing on the intervals (1, 3) and (3, ∞).

Derivative and Critical Points – Understanding how to find the derivative of a function to determine critical points and analyze intervals of increase and decrease.
Increasing and Decreasing Intervals – Identifying where the function is increasing based on the sign of the derivative.