use the given information to sketch the graph of f.\ndomain: all real x, except x = - 2 and x =…

use the given information to sketch the graph of f.\ndomain: all real x, except x = - 2 and x = 2.\nf(-3)=-2, f(0)=0; f(3)=2.\nf(x)<0 on (-∞,-2) and (2,∞); f(x)>0 on (-2,2).\nf(x)<0 on (-∞,-2) and (-2,0); f(x)>0 on (0,2) and (2,∞).\nvertical asymptotes: x = - 2 and x = 2. horizontal asymptote: y = 0.\nchoose the correct graph below.\n

Answer

Explanation:

Step1: Analyze the function's behavior based on the first - derivative

Since (f^{\prime}(x)<0) on ((-\infty,-2)) and ((2,\infty)), the function is decreasing on these intervals.
Since (f^{\prime}(x)>0) on ((-2,2)), the function is increasing on this interval.

Step2: Analyze the function's concavity based on the second - derivative

Since (f^{\prime\prime}(x)<0) on ((-\infty,-2)) and ((-2,0)), the function is concave down on these intervals.
Since (f^{\prime\prime}(x)>0) on ((0,2)) and ((2,\infty)), the function is concave up on these intervals.

Step3: Consider the asymptotes

Vertical asymptotes at (x = - 2) and (x = 2).
Horizontal asymptote at (y = 0).

Step4: Check the function values

(f(-3)=-2), (f(0) = 0), (f(3)=2).

Now, let's analyze each option:

Option A: Does not follow the concavity and asymptote rules correctly.
Option B: Follows the rules of increasing/decreasing (from the first - derivative), concavity (from the second - derivative), and asymptotes.
Option C: Does not follow the concavity and asymptote rules correctly.