Function \( f(x)=\displaystyle\frac{3}{2}x^{4}-4x^{3}-45x^{2}+25 \) is :
(A) increasing in \( (-\infty,-3) \)
(B) Decreasing in \( (-3,0) \)
(C) increasing in \( (-3,0) \)
(D) Decreasing in \( (0,5) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (C) and (D) only
The sign of the derivative decides where \( f \) increases or decreases:
\( \displaystyle f'(x)=6x^{3}-12x^{2}-90x \)
\( \displaystyle =6x(x+3)(x-5) \)
The zeros of \( f' \) are \( -3,\ 0,\ 5 \). \( f'(x) \) is negative for \( x\lt -3 \), positive on \( (-3,0) \), negative on \( (0,5) \) and positive for \( x\gt 5 \).
So \( f \) is decreasing on \( (-\infty,-3) \), increasing on \( (-3,0) \), decreasing on \( (0,5) \) and increasing on \( (5,\infty) \).
Checking the statements
(A) increasing in \( (-\infty,-3) \) — Incorrect. \( f'\lt 0 \) there, so \( f \) is decreasing.
(B) Decreasing in \( (-3,0) \) — Incorrect. \( f'\gt 0 \) there, so \( f \) is increasing.
(C) increasing in \( (-3,0) \) — Correct. \( f'\gt 0 \) on \( (-3,0) \).
(D) Decreasing in \( (0,5) \) — Correct. \( f'\lt 0 \) on \( (0,5) \).