Answer :
Let's analyze the function [tex]\( f(x) = -(x + 6)(x + 2) \)[/tex] step-by-step to determine which statements about this function are true.
### 1. Finding the First Derivative
To understand the intervals where the function is increasing or decreasing, we first need to find the critical points by computing the first derivative [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = -(x + 6)(x + 2) \][/tex]
Using the product rule and chain rule:
[tex]\[ f'(x) = -[(x + 6)'(x + 2) + (x + 6)(x + 2)'] \][/tex]
[tex]\[ f'(x) = -[(1)(x + 2) + (x + 6)(1)] \][/tex]
[tex]\[ f'(x) = -(x + 2 + x + 6) \][/tex]
[tex]\[ f'(x) = -(2x + 8) \][/tex]
[tex]\[ f'(x) = -2(x + 4) \][/tex]
### 2. Finding the Critical Points
Next, we set the first derivative equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -2(x + 4) = 0 \][/tex]
[tex]\[ x + 4 = 0 \][/tex]
[tex]\[ x = -4 \][/tex]
So, the critical point is at [tex]\( x = -4 \)[/tex].
### 3. Analyzing the Intervals Around the Critical Points
We need to determine the sign of [tex]\( f'(x) \)[/tex] in the intervals around the critical point to identify where the function is increasing or decreasing. The critical point divides the real number line into two intervals: [tex]\( x < -4 \)[/tex] and [tex]\( x > -4 \)[/tex].
#### Interval 1: [tex]\( x < -4 \)[/tex]
Choose a test point in this interval, say [tex]\( x = -5 \)[/tex]:
[tex]\[ f'(-5) = -2(-5 + 4) = -2(-1) = 2 \][/tex]
Since [tex]\( f'(-5) > 0 \)[/tex], [tex]\( f(x) \)[/tex] is increasing in the interval [tex]\( x < -4 \)[/tex].
#### Interval 2: [tex]\( x > -4 \)[/tex]
Choose a test point in this interval, say [tex]\( x = -3 \)[/tex]:
[tex]\[ f'(-3) = -2(-3 + 4) = -2(1) = -2 \][/tex]
Since [tex]\( f'(-3) < 0 \)[/tex], [tex]\( f(x) \)[/tex] is decreasing in the interval [tex]\( x > -4 \)[/tex].
### 4. Conclusion
Based on the analysis above, we conclude:
- The function [tex]\( f(x) \)[/tex] is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].
- The function [tex]\( f(x) \)[/tex] is decreasing for all real values of [tex]\( x \)[/tex] where [tex]\( x > -4 \)[/tex].
Thus, the correct statement about the function is:
The function is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].
### 1. Finding the First Derivative
To understand the intervals where the function is increasing or decreasing, we first need to find the critical points by computing the first derivative [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = -(x + 6)(x + 2) \][/tex]
Using the product rule and chain rule:
[tex]\[ f'(x) = -[(x + 6)'(x + 2) + (x + 6)(x + 2)'] \][/tex]
[tex]\[ f'(x) = -[(1)(x + 2) + (x + 6)(1)] \][/tex]
[tex]\[ f'(x) = -(x + 2 + x + 6) \][/tex]
[tex]\[ f'(x) = -(2x + 8) \][/tex]
[tex]\[ f'(x) = -2(x + 4) \][/tex]
### 2. Finding the Critical Points
Next, we set the first derivative equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -2(x + 4) = 0 \][/tex]
[tex]\[ x + 4 = 0 \][/tex]
[tex]\[ x = -4 \][/tex]
So, the critical point is at [tex]\( x = -4 \)[/tex].
### 3. Analyzing the Intervals Around the Critical Points
We need to determine the sign of [tex]\( f'(x) \)[/tex] in the intervals around the critical point to identify where the function is increasing or decreasing. The critical point divides the real number line into two intervals: [tex]\( x < -4 \)[/tex] and [tex]\( x > -4 \)[/tex].
#### Interval 1: [tex]\( x < -4 \)[/tex]
Choose a test point in this interval, say [tex]\( x = -5 \)[/tex]:
[tex]\[ f'(-5) = -2(-5 + 4) = -2(-1) = 2 \][/tex]
Since [tex]\( f'(-5) > 0 \)[/tex], [tex]\( f(x) \)[/tex] is increasing in the interval [tex]\( x < -4 \)[/tex].
#### Interval 2: [tex]\( x > -4 \)[/tex]
Choose a test point in this interval, say [tex]\( x = -3 \)[/tex]:
[tex]\[ f'(-3) = -2(-3 + 4) = -2(1) = -2 \][/tex]
Since [tex]\( f'(-3) < 0 \)[/tex], [tex]\( f(x) \)[/tex] is decreasing in the interval [tex]\( x > -4 \)[/tex].
### 4. Conclusion
Based on the analysis above, we conclude:
- The function [tex]\( f(x) \)[/tex] is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].
- The function [tex]\( f(x) \)[/tex] is decreasing for all real values of [tex]\( x \)[/tex] where [tex]\( x > -4 \)[/tex].
Thus, the correct statement about the function is:
The function is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].
