Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

The graph of the function [tex][tex]$f(x)=-(x+6)(x+2)$[/tex][/tex] is shown below.

Which statement about the function is true?

A. The function is increasing for all real values of [tex][tex]$x$[/tex][/tex] where [tex][tex]$x\ \textless \ -4$[/tex][/tex].
B. The function is increasing for all real values of [tex][tex]$x$[/tex][/tex] where [tex][tex]$-6\ \textless \ x\ \textless \ -2$[/tex][/tex].
C. The function is decreasing for all real values of [tex][tex]$x$[/tex][/tex] where [tex][tex]$x\ \textless \ -6$[/tex][/tex] and where [tex][tex]$x\ \textgreater \ -2$[/tex][/tex].
D. The function is decreasing for all real values of [tex][tex]$x$[/tex][/tex] where [tex][tex]$x\ \textless \ -4$[/tex][/tex].

Sagot :

GinnyAnswer
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].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.