Question

use the given information to sketch the graph of f.
domain: all real x, except x = - 2 and x = 2.
f(-3)=-2, f(0)=0; f(3)=2.
f(x)<0 on (-∞,-2) and (2,∞); f(x)>0 on (-2,2).
f(x)<0 on (-∞,-2) and (-2,0); f(x)>0 on (0,2) and (2,∞).
vertical asymptotes: x = - 2 and x = 2. horizontal asymptote: y = 0.
choose the correct graph below.

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.

Answer:

B.