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Sagot :
To determine which statement about the function [tex]\( f(x) = (x+2)(x+6) \)[/tex] is true, let's first identify the roots and the behavior of the function around these points.
1. Finding the roots of the function:
The roots of the function occur where [tex]\( f(x) = 0 \)[/tex]. By setting [tex]\( (x+2)(x+6) = 0 \)[/tex] and solving for [tex]\( x \)[/tex]:
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
[tex]\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \][/tex]
So, the roots are [tex]\( x = -2 \)[/tex] and [tex]\( x = -6 \)[/tex].
2. Determining the sign of the function in different intervals:
The function [tex]\( f(x) \)[/tex] changes its sign at the roots [tex]\( x = -2 \)[/tex] and [tex]\( x = -6 \)[/tex]. We will test the sign of the function in the intervals:
- [tex]\( x < -6 \)[/tex]
- [tex]\( -6 < x < -2 \)[/tex]
- [tex]\( x > -2 \)[/tex]
3. Evaluating the function at test points in each interval:
- For [tex]\( x < -6 \)[/tex], choose [tex]\( x = -7 \)[/tex]:
[tex]\[ f(-7) = (-7 + 2)(-7 + 6) = (-5)(-1) = 5 \][/tex]
The function is positive in this interval.
- For [tex]\( -6 < x < -2 \)[/tex], choose [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4 \][/tex]
The function is negative in this interval.
- For [tex]\( x > -2 \)[/tex], choose [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = (-1 + 2)(-1 + 6) = (1)(5) = 5 \][/tex]
The function is positive in this interval.
4. Summary of the function's behavior:
- [tex]\( f(x) > 0 \)[/tex] when [tex]\( x < -6 \)[/tex] or [tex]\( x > -2 \)[/tex]
- [tex]\( f(x) < 0 \)[/tex] when [tex]\( -6 < x < -2 \)[/tex]
Now, let's evaluate each statement:
1. The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x > -4 \)[/tex]:
This is incorrect because the function is negative in the interval [tex]\( -6 < x < -2 \)[/tex].
2. The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex]:
This is correct based on our evaluation.
3. The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x < -6 \)[/tex] or [tex]\( x > -3 \)[/tex]:
This is incorrect because the interval [tex]\( x > -3 \)[/tex] does not accurately capture the positive intervals (which should be [tex]\( x > -2 \)[/tex]).
4. The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( x < -2 \)[/tex]:
This is incorrect because the function is positive when [tex]\( x < -6 \)[/tex].
Therefore, the correct statement is:
The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].
1. Finding the roots of the function:
The roots of the function occur where [tex]\( f(x) = 0 \)[/tex]. By setting [tex]\( (x+2)(x+6) = 0 \)[/tex] and solving for [tex]\( x \)[/tex]:
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
[tex]\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \][/tex]
So, the roots are [tex]\( x = -2 \)[/tex] and [tex]\( x = -6 \)[/tex].
2. Determining the sign of the function in different intervals:
The function [tex]\( f(x) \)[/tex] changes its sign at the roots [tex]\( x = -2 \)[/tex] and [tex]\( x = -6 \)[/tex]. We will test the sign of the function in the intervals:
- [tex]\( x < -6 \)[/tex]
- [tex]\( -6 < x < -2 \)[/tex]
- [tex]\( x > -2 \)[/tex]
3. Evaluating the function at test points in each interval:
- For [tex]\( x < -6 \)[/tex], choose [tex]\( x = -7 \)[/tex]:
[tex]\[ f(-7) = (-7 + 2)(-7 + 6) = (-5)(-1) = 5 \][/tex]
The function is positive in this interval.
- For [tex]\( -6 < x < -2 \)[/tex], choose [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4 \][/tex]
The function is negative in this interval.
- For [tex]\( x > -2 \)[/tex], choose [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = (-1 + 2)(-1 + 6) = (1)(5) = 5 \][/tex]
The function is positive in this interval.
4. Summary of the function's behavior:
- [tex]\( f(x) > 0 \)[/tex] when [tex]\( x < -6 \)[/tex] or [tex]\( x > -2 \)[/tex]
- [tex]\( f(x) < 0 \)[/tex] when [tex]\( -6 < x < -2 \)[/tex]
Now, let's evaluate each statement:
1. The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x > -4 \)[/tex]:
This is incorrect because the function is negative in the interval [tex]\( -6 < x < -2 \)[/tex].
2. The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex]:
This is correct based on our evaluation.
3. The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x < -6 \)[/tex] or [tex]\( x > -3 \)[/tex]:
This is incorrect because the interval [tex]\( x > -3 \)[/tex] does not accurately capture the positive intervals (which should be [tex]\( x > -2 \)[/tex]).
4. The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( x < -2 \)[/tex]:
This is incorrect because the function is positive when [tex]\( x < -6 \)[/tex].
Therefore, the correct statement is:
The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].
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