For the function \( f(x)=\displaystyle\frac{3}{2}x^4-4x^3-45x^2+49 \). Which of the following statements is/are TRUE ?
A. \( f(x) \) is increasing in \( (-\infty,-3) \)
B. \( f(x) \) is increasing in \( (-3,0)\cup(5,\infty) \)
C. \( f(x) \) is decreasing in \( (-\infty,-3)\cup(0,5) \)
D. \( f(x) \) is decreasing in \( (-\infty,-3)\cup(5,\infty) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B and C only
\( \displaystyle f'(x)=6x^3-12x^2-90x \)
\( \displaystyle =6x(x^2-2x-15) \)
\( \displaystyle =6x(x+3)(x-5) \)
Signs of the factors x, (x + 3), (x − 5) on each interval:
On \( (-\infty,-3) \): (−)(−)(−), so \( f'(x)\lt 0 \)
On (−3, 0): (−)(+)(−), so \( f'(x)\gt 0 \)
On (0, 5): (+)(+)(−), so \( f'(x)\lt 0 \)
On \( (5,\infty) \): (+)(+)(+), so \( f'(x)\gt 0 \)
A statement such as “increasing in \( (-3,0)\cup(5,\infty) \)” is read, as usual, as increasing on each of the two intervals.
So f is decreasing on \( (-\infty,-3) \) and (0, 5), and increasing on (−3, 0) and \( (5,\infty) \).
Checking the statements
(A) \( f(x) \) is increasing in \( (-\infty,-3) \) — Incorrect. \( f'(x)\lt 0 \) there, so f is decreasing.
(B) \( f(x) \) is increasing in \( (-3,0)\cup(5,\infty) \) — Correct. \( f'(x)\gt 0 \) on both intervals.
(C) \( f(x) \) is decreasing in \( (-\infty,-3)\cup(0,5) \) — Correct. \( f'(x)\lt 0 \) on both intervals.
(D) \( f(x) \) is decreasing in \( (-\infty,-3)\cup(5,\infty) \) — Incorrect. \( f'(x)\gt 0 \) on \( (5,\infty) \), so f is increasing there.