For the function, \(f(x) = \displaystyle\frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350\), 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:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
The critical points are where \(f'(x) = 0\):
\( \displaystyle f'(x) = -3x^3 - 24x^2 - 45x \)
\( \displaystyle = -3x(x^2 + 8x + 15) \)
\( \displaystyle = -3x(x + 3)(x + 5) \)
For the second derivative test:
\( \displaystyle f''(x) = -9x^2 - 48x - 45 \)
So the critical points are x = 0, −3 and −5.
Checking the statements
(A) \(x = -3\) and \(x = -5\) are the only critical points of the given function. — Incorrect. x = 0 is also a critical point.
(B) \(x = -3\) is a point of local minimum. — Correct. \(f''(-3) = -81 + 144 - 45 = 18 \gt 0\).
(C) The local minimum value at \(x = -3\) is 231. — Incorrect. The local minimum value is −397.25, not 231:
\( \displaystyle f(-3) = -\frac{3}{4}(81) - 8(-27) - \frac{45}{2}(9) - 350 \)
\( \displaystyle = -60.75 + 216 - 202.5 - 350 \)
\( \displaystyle = -397.25 \)
(D) \(x = -5\) is a point of local maximum. — Correct. \(f''(-5) = -225 + 240 - 45 = -30 \lt 0\).