Final answer:
To find intervals where the function is increasing or decreasing, we calculate its first derivative and set it to zero to find the critical points. We find that f(x) is decreasing on (0, 4) and increasing on (4, 20).
Step-by-step explanation:
To find the intervals on which the function f(x) = 2x³ + 3x² − 120x is increasing or decreasing, we need to determine its first derivative, f'(x), and analyze its sign. We find the first derivative of the function:
f'(x) = 6x² + 6x - 120.
Next, we look for the critical points by setting f'(x) equal to zero and solving for x:
- 6x² + 6x - 120 = 0
- Divide through by 6: x² + x - 20 = 0
- Factor the quadratic: (x - 4)(x + 5) = 0
- Solve for x: x = 4 or x = -5
Since the question asks for intervals between 0 and 20, we disregard the negative value and use only x = 4. Now, we test values on either side of x = 4 to determine where f'(x) is positive (function increasing) or negative (function decreasing).
- For x < 4, let x = 0: f'(0) = -120, which is negative.
- For x > 4, let x = 5: f'(5) = 6(25) + 6(5) - 120 = 150 + 30 - 120, which is positive.
Therefore, the function is decreasing on the interval (0, 4) and increasing on the interval (4, 20).