You are told that $f(x)$ has a local max at $x=7$ and neither a local max nor a local min at $x=2$.
Which of the following statements are consistent with this information?
Select all that apply.
Correct answers are worth 0.5 marks each, to a max of 1 mark on this question.
Incorrect answers are worth -0.5 marks each, to a min of 0 marks on this question.
A. $f^{\prime}(7)=0$ and $f^{\prime \prime}(7)=-5$
B. $f^{\prime}(2)=3$ and $f^{\prime \prime}(2)=0$
C. $f^{\prime}(2)=0$ and $f^{\prime \prime}(2)=-3$
D. $f^{\prime}(7)=5$ and $f^{\prime \prime}(7)=5$
E. $f^{\prime}(7)=-5$ and $f^{\prime \prime}(7)=-5$
F. $f^{\prime}(2)=0$ and $f^{\prime \prime}(2)=3$
G. $f^{\prime}(7)=0$ and $f^{\prime \prime}(7)=5$
H. $f^{\prime}(7)=-5$ and $f^{\prime \prime}(7)=0$

Step 1 ：Given that $f(x)$ has a local max at $x=7$ and neither a local max nor a local min at $x=2$.

Step 2 ：From the first derivative test, we know that if $f(x)$ has a local max at $x=a$, then $f^{\prime}(a)=0$.

Step 3 ：From the second derivative test, we know that if $f(x)$ has a local max at $x=a$, then $f^{\prime \prime}(a)<0$.

Step 4 ：So, the statement $f^{\prime}(7)=0$ and $f^{\prime \prime}(7)=-5$ is consistent with the given information.

Step 5 ：Since $f(x)$ has neither a local max nor a local min at $x=2$, $f^{\prime}(2)$ can be any real number except 0.

Step 6 ：Also, $f^{\prime \prime}(2)$ can be any real number.

Step 7 ：So, the statement $f^{\prime}(2)=3$ and $f^{\prime \prime}(2)=0$ is consistent with the given information.

Step 8 ：Final Answer: The statements that are consistent with the given information are \(\boxed{A}\) and \(\boxed{B}\).