For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals of increase. (2022

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Question: For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals of increase. (2022)

Options:

Correct Answer: (0, 3)

Exam Year: 2022

Solution:

f\'(x) = 6x^2 - 18x + 12. The critical points are x = 1 and x = 2. Test intervals show increase in (0, 3).

For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals of increase. (2022)

For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals of increase. (2022)

Step 1: Start with the function f(x) = 2x^3 - 9x^2 + 12x.
Step 2: Find the derivative of the function, f'(x). This tells us the slope of the function.
Step 3: Calculate the derivative: f'(x) = 6x^2 - 18x + 12.
Step 4: Set the derivative equal to zero to find critical points: 6x^2 - 18x + 12 = 0.
Step 5: Simplify the equation by dividing everything by 6: x^2 - 3x + 2 = 0.
Step 6: Factor the quadratic equation: (x - 1)(x - 2) = 0.
Step 7: Solve for x to find critical points: x = 1 and x = 2.
Step 8: Identify the intervals to test: (-∞, 1), (1, 2), and (2, ∞).
Step 9: Choose a test point in each interval and plug it into f'(x) to see if it's positive (increasing) or negative (decreasing).
Step 10: For the interval (-∞, 1), test x = 0: f'(0) = 12 (positive, so increasing).
Step 11: For the interval (1, 2), test x = 1.5: f'(1.5) = -1.5 (negative, so decreasing).
Step 12: For the interval (2, ∞), test x = 3: f'(3) = 12 (positive, so increasing).
Step 13: Combine the results: The function is increasing on the intervals (0, 1) and (2, ∞).

Derivative and Critical Points – Understanding how to find the derivative of a function to identify critical points where the function may change from increasing to decreasing or vice versa.
Test Intervals – Using test intervals around critical points to determine where the function is increasing or decreasing.
Function Behavior – Analyzing the behavior of polynomial functions to determine intervals of increase and decrease.