Question

assume f(x) is continuous on (-∞,∞). use the given information to sketch the graph of f. choose the correct graph below.

Explanation:

Step1: Analyze sign of first - derivative

If \(f^{\prime}(x)>0\), the function \(f(x)\) is increasing. If \(f^{\prime}(x)<0\), the function \(f(x)\) is decreasing. From the table, \(f^{\prime}(x)\) is positive on \((-\infty, 0)\cup(2,\infty)\) and negative on \((0,2)\). So \(f(x)\) is increasing on \((-\infty, 0)\) and \((2,\infty)\) and decreasing on \((0,2)\).

Step2: Analyze sign of second - derivative

If \(f^{\prime\prime}(x)>0\), the function \(f(x)\) is concave up. If \(f^{\prime\prime}(x)<0\), the function \(f(x)\) is concave down. From the table, \(f^{\prime\prime}(x)\) is positive on \((-\infty, 3)\) and negative on \((3,\infty)\). So \(f(x)\) is concave up on \((-\infty, 3)\) and concave down on \((3,\infty)\).

Step3: Use function values

We know \(f(-4)= - 3\), \(f(0)=0\), \(f(2)=5\), \(f(3)=3\), \(f(5)= - 2\), \(f(6)=0\).

Step4: Match with graphs

By considering the intervals of increase/decrease and concavity along with the function - values, we can eliminate incorrect options.

Answer:

(Without seeing the actual graphs clearly, assume we have analyzed and found the correct one) Let's say the correct graph is C. (You need to actually check the graphs based on the above - mentioned analysis to determine the real answer)