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

Which statement about the function is true?

A. The function is increasing for all real values of [tex]x[/tex] where [tex]x\ \textless \ -4[/tex].

B. The function is increasing for all real values of [tex]x[/tex] where [tex]-6\ \textless \ x\ \textless \ -2[/tex].

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

D. The function is decreasing for all real values of [tex]x[/tex] where [tex]x\ \textless \ -4[/tex].



Answer :

To determine which statement about the function [tex]\( f(x) = -(x+6)(x+2) \)[/tex] is true, let's analyze the function step by step:

1. Expand the function:

[tex]\( f(x) = -(x+6)(x+2) \)[/tex]

Expanding the product inside the parentheses:

[tex]\( f(x) = -(x^2 + 8x + 12) \)[/tex]

Simplify by distributing the negative sign:

[tex]\( f(x) = -x^2 - 8x - 12 \)[/tex]

2. Find the derivative of the function to determine its critical points and behavior:

The derivative [tex]\( f'(x) \)[/tex] is calculated as follows:

[tex]\( f'(x) = \frac{d}{dx} (-x^2 - 8x - 12) \)[/tex]

Using the power rule:

[tex]\( f'(x) = -2x - 8 \)[/tex]

3. Set the derivative equal to zero to find critical points:

[tex]\( -2x - 8 = 0 \)[/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\( -2x = 8 \)[/tex]

[tex]\( x = -4 \)[/tex]

The critical point is [tex]\( x = -4 \)[/tex].

4. Determine whether the function is increasing or decreasing around the critical point:

- For [tex]\( x < -4 \)[/tex], let's choose [tex]\( x = -5 \)[/tex]:

[tex]\[ f'(-5) = -2(-5) - 8 = 10 - 8 = 2 \][/tex]

Since [tex]\( f'(-5) > 0 \)[/tex], the function is increasing for [tex]\( x < -4 \)[/tex].

- For [tex]\( x > -4 \)[/tex], let's choose [tex]\( x = -3 \)[/tex]:

[tex]\[ f'(-3) = -2(-3) - 8 = 6 - 8 = -2 \][/tex]

Since [tex]\( f'(-3) < 0 \)[/tex], the function is decreasing for [tex]\( x > -4 \)[/tex].

5. Evaluate the given statements:

- The function is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].

This is true because we determined that the derivative [tex]\( f'(x) \)[/tex] is positive for [tex]\( x < -4 \)[/tex].

- The function is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].

This is false because the function is not increasing in the interval [tex]\( -6 < x < -2 \)[/tex], especially since it decreases for [tex]\( x > -4 \)[/tex].

- The function is decreasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -6 \)[/tex] and where [tex]\( x > -2 \)[/tex].

This is false because we have established that the function is increasing for [tex]\( x < -4 \)[/tex].

- The function is decreasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].

This is false as the function is increasing in that interval.

Therefore, the correct statement is:

The function is increasing for all real values of [tex]\( x \)[/tex] where [tex]\( x < -4 \)[/tex].

Thus, the correct answer is the first statement.