High School

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Find the intervals on which \( f \) is increasing and the intervals on which it is decreasing.

Given:
\[ f(x) = -12x^5 + 60x^4 - 60x^3 \]

Find \( f'(x) \).

Answer :

Final answer:

f(x) is increasing on the intervals (-∞, 0) and (1, 3), and it is decreasing on the intervals (0, 1) and (3, ∞) after finding the critical points and testing the sign of f'(x) in the relevant intervals.

Explanation:

To find the intervals on which the given function f(x) = -12x5 + 60x4 - 60x3 is increasing or decreasing, we must first calculate its first derivative, f'(x). The derivative of f(x) is f'(x) = -60x4 + 240x3 - 180x2.

Next, we determine the critical points where f'(x) = 0:

  • Solve -60x4 + 240x3 - 180x2 = 0 for x.
  • Factor the equation: -60x2(x2 - 4x + 3) = 0.
  • Find roots: x = 0, 1, 3.

Then we use these points to test the sign of f'(x) in the intervals to determine where the function is increasing or decreasing:

  1. Interval (-∞, 0): Choose x = -1, f'(-1) > 0, so f is increasing.
  2. Interval (0, 1): Choose x = 0.5, f'(0.5) < 0, so f is decreasing.
  3. Interval (1, 3): Choose x = 2, f'(2) > 0, so f is increasing.
  4. Interval (3, ∞): Choose x = 4, f'(4) < 0, so f is decreasing.

Therefore, the function is increasing on the intervals (-∞, 0) and (1, 3) and decreasing on the intervals (0, 1) and (3, ∞).

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