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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].
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