Question

The function f(x) = 2x³ – 9x² + 12x +29 is monotonically increasing in the interval.

• A. (–∞,∞)
• B. (–∞,1) U (2, ∞)
• C. (–∞,1)
• D. <div>(2, ∞)</div>

Answer 1

The Correct Option is A

To determine the intervals in which the function f(x) = 2x³ - 9x² + 12x + 29 is monotonically increasing, we need to analyze its derivative. 
find the derivative of f(x):
f'(x) = \(\frac {d}{dx}\) (2x³ - 9x² + 12x + 29)= 6x² - 18x + 12.
To determine where the derivative is positive (indicating increasing behavior), we need to find the roots of f'(x) = 0. So, 6x² - 18x + 12 = 0.
Dividing by 6;
x² - 3x + 2 = 0.
Factoring; 
(x - 1)(x - 2) = 0
So, the roots are x = 1 and x = 2.
Now, we can create a number line and test the sign of f'(x) in each interval:Interval (-∞, 1):
Choose a test point x = 0:
f'(0) = 6(0)² - 18(0) + 12 = 12.
Since f'(0) > 0, f(x) is increasing in the interval (-∞, 1).Interval (1, 2):
Choose a test point x = 1.5:
f'(1.5) = 6(1.5)² - 18(1.5) + 12 = -9
Since f'(1.5) < 0, f(x) is not increasing in the interval (1, 2).Interval (2, ∞):
Choose a test point x = 3:
f'(3) = 6(3)² - 18(3) + 12 = 0.
Since f'(3) = 0, we cannot determine the behavior of f(x) in this interval. 
we can conclude that f(x) is monotonically increasing in the interval (-∞, 1) and monotonically decreasing in the interval (1, 2). Therefore, the correct answer is option (C) (-∞,1)

Answer 2

To find where the function f(x) = 2x³ - 9x² + 12x + 29 is monotonically increasing, we analyze its derivative:
1. Find the Derivative:
f'(x) = 6x² - 18x + 12
2. Set the Derivative Equal to Zero to Find Critical Points:
6x² - 18x + 12 = 0
3. Solve for x:
x² - 3x + 2 = 0
(x - 1)(x - 2) = 0
x = 1 or x = 2
4. Test Intervals Using the Number Line Method:
 
Interval (-∞, 1): Test with x = 0 
f'(0) = 12 (positive, so f(x) is increasing)
Interval (1, 2): Test with x = 1.5
f'(1.5) = -9 (negative, so f(x) is not increasing)
Interval (2, ∞): Test with x = 3
f'(3) = 0 (no conclusion about the trend of f(x))

5. Conclusion:
- f(x) is monotonically increasing in the interval (-∞, 1).
- f(x) is not increasing (and thus either decreasing or constant) in the interval (1, 2).