Suppose $f(x)$ has critical points at $x=2, x=1$, and $x=-1$ and that $f^{\prime \prime}(x)=\frac{2 x^{3}+7 x^{2}-12 x-5}{(3+x)^{2}}$.
Which of the following statements are true? Select all that apply.
Correct answers are worth 0.33 marks, to a max of 1 mark on this question.
Incorrect answers are worth -0.33 marks, to a min of 0 on this question.
A. There is a local max at $x=1$
B. There is a local min at $x=2$
C. There is a local max at $x=2$
D. None of these points are locations of local max or local min.
E. There is a local max at $x=-1$
F. There is a local $\min$ at $x=-1$
G. There is a local min at $x=1$

Step 1 ：Given that $f(x)$ has critical points at $x=2, x=1$, and $x=-1$ and that $f^{\prime \prime}(x)=\frac{2 x^{3}+7 x^{2}-12 x-5}{(3+x)^{2}}$.

Step 2 ：We can use the second derivative test to determine whether a critical point is a local maximum, minimum, or neither. The second derivative test states that if the second derivative at a point is positive, then the function has a local minimum at that point. If the second derivative at a point is negative, then the function has a local maximum at that point. If the second derivative at a point is zero, then the test is inconclusive.

Step 3 ：Calculate the second derivative at $x=2$, $x=1$, and $x=-1$ respectively.

Step 4 ：The second derivative at $x=2$ is \(\frac{3}{5}\), which is positive, indicating a local minimum at $x=2$.

Step 5 ：The second derivative at $x=1$ is \(-\frac{1}{2}\), which is negative, indicating a local maximum at $x=1$.

Step 6 ：The second derivative at $x=-1$ is 3, which is positive, indicating a local minimum at $x=-1$.

Step 7 ：Thus, the statements B, A, and F are true.

Step 8 ：Final Answer: \(\boxed{\text{B. There is a local min at } x=2, \text{A. There is a local max at } x=1, \text{F. There is a local min at } x=-1}\)