For the function, $f(x) =\frac{-3}{4}x^4-8x^3-\frac{45}{2}x^2-350$, which of the following statements are correct?

(A) $x=-3$ and $x = -5$ are the only critical points of the given function.
(B) $x=-3$ is a point of local minimum.
(C) The local minimum value at $x = -3$ is 231.
(D) $x = -5$ is a point of local maximum.

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (2) → (B) and (D) only

Given: $f(x) = -\frac{3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$

$f'(x) = -3x^3 - 24x^2 - 45x = -3x(x+3)(x+5)$

Critical points: $x = 0,-3,-5$ ⇒ (A) is false.

$f''(x) = -9x^2 - 48x - 45$

At $x=-3$: $f''(-3) = 18 > 0$ ⇒ local minimum at $x=-3$ ⇒ (B) true.

$f(-3) = -\frac{3}{4}\cdot 81 + 216 - \frac{45}{2}\cdot 9 - 350 = -397.25 \ne 231$ ⇒ (C) false.

At $x=-5$: $f''(-5) = -30 < 0$ ⇒ local maximum at $x=-5$ ⇒ (D) true.