Question

Let $ f(x) = x^3 - 6x^2 + 9x + 8 $, then $ f(x) $ is decreasing in

• A. \( (-\infty, 1) \)
• B. \( [1, 3] \)
• C. \( [3, \infty) \)
• D. \( (-\infty, 1) \cup (3, \infty) \)

Hint:

To determine where a function is increasing or decreasing, find the critical points by setting the derivative equal to zero, and then test the sign of the derivative in the intervals.

Answer 1

The Correct Option is A

Step 1: Find the derivative of \( f(x) \)
The first derivative of \( f(x) \) is: \[ f'(x) = 3x^2 - 12x + 9 \]
Step 2: Solve for Critical Points
Set the derivative equal to zero to find the critical points: \[ 3x^2 - 12x + 9 = 0 \] Simplifying: \[ x^2 - 4x + 3 = 0 \] Factoring: \[ (x - 1)(x - 3) = 0 \] Thus, the critical points are \( x = 1 \) and \( x = 3 \).
Step 3: Determine the intervals of increase and decrease
Using a sign chart, test the intervals around the critical points \( x = 1 \) and \( x = 3 \):
For \( x < 1 \), \( f'(x) > 0 \) (function is increasing)
For \( 1 < x < 3 \), \( f'(x) < 0 \) (function is decreasing)
For \( x > 3 \), \( f'(x) > 0 \) (function is increasing)

Step 4: Conclusion
Thus, \( f(x) \) is decreasing in the interval \( (-\infty, 1) \).