Determine the location of each local extremum of the function.
\[
f(x)=-x^{3}-9 x^{2}-15 x-2
\]

What is/are the local minimum/minima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The local minimum/minima is/are -27 at $\mathrm{x}=-5$.
(Use a comma to separate answers as needed. Type integers or simplified fractions.)
B. The function has no local minimum.

What is/are the local maximum/maxima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The local maximum/maxima is/are $\square$ at $x=\square$.
(Use a comma to separate answers as needed. Type integers or simplified fractions.)
B. The function has no local maximum.

Step 1 ：The function is \(f(x)=-x^{3}-9 x^{2}-15 x-2\).

Step 2 ：The derivative of the function is \(f'(x)=-3x^{2}-18x-15\).

Step 3 ：Setting the derivative equal to zero gives us the equation \(-3x^{2}-18x-15=0\).

Step 4 ：We can factor out a -3 to simplify the equation: \(x^{2}+6x+5=0\).

Step 5 ：Factoring the quadratic gives us \((x+1)(x+5)=0\).

Step 6 ：Setting each factor equal to zero gives us the critical points \(x=-1\) and \(x=-5\).

Step 7 ：The second derivative of the function is \(f''(x)=-6x-18\).

Step 8 ：Substituting \(x=-1\) into the second derivative gives us \(f''(-1)=-6(-1)-18=6-18=-12\), which is less than zero, so \(x=-1\) is a local maximum.

Step 9 ：Substituting \(x=-5\) into the second derivative gives us \(f''(-5)=-6(-5)-18=30-18=12\), which is greater than zero, so \(x=-5\) is a local minimum.

Step 10 ：For \(x=-1\), \(f(-1)=-(-1)^{3}-9(-1)^{2}-15(-1)-2=1-9+15-2=5\).

Step 11 ：For \(x=-5\), \(f(-5)=-(-5)^{3}-9(-5)^{2}-15(-5)-2=-125-225+75-2=-277\).

Step 12 ：\(\boxed{\text{So, the local minimum is -277 at } x=-5 \text{ and the local maximum is 5 at } x=-1}\).