Let $$f(x) = x^3 + 15x^2 - 33x - 36$$ be a real-valued function. Which of the following statements is/are TRUE?

To determine the local extrema of the function $f(x) = x^3 + 15x^2 - 33x - 36$, find its first derivative:

Set $f^{\prime}(x) = 0$ to find the critical points:

Thus, the critical points are $x = -11$ and $x = 1$.

Now, compute the second derivative:

Evaluate the second derivative at the critical points:

1. At $x = -11$:

   $$f^{\prime\prime}(-11) = 6(-11) + 30 = -36 < 0$$

   Since $f''(-11) < 0$, $f(x)$ has alocal maximum at $x = -11$.

2. At $x = 1$:

   $$f^{\prime\prime}(1) = 6(1) + 30 = 36 > 0$$

   Since $f''(1) > 0$, $f(x)$ has alocal minimum at $x = 1$.

Therefore, statements (B) and (D) are TRUE, while (A) and (C) are FALSE.