For the function \( f(x)=x^{4}-2x^{2}+5,x\in R \), then which of the following statements are True?
A. \( f(x) \) is increasing on \( (-\infty,-1) \)
B. \( f(x) \) is decreasing on \( (-\infty,-1) \)
C. \( f(x) \) is increasing on \( (-1,0)\cup(1,\infty) \)
D. \( f(x) \) is decreasing on \( (1,\infty) \)
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
\( \displaystyle f'(x)=4x^{3}-4x \)
\( \displaystyle =4x(x-1)(x+1) \)
The critical points −1, 0, 1 split the line into four intervals. The sign of f′(x) on each:
\( \displaystyle x\lt -1:\ f'(x)\lt 0 \)
\( \displaystyle -1\lt x\lt 0:\ f'(x)\gt 0 \)
\( \displaystyle 0\lt x\lt 1:\ f'(x)\lt 0 \)
\( \displaystyle x\gt 1:\ f'(x)\gt 0 \)
(for example f′(−2) = −24 and f′(−0.5) = 1.5). In (C), “increasing on \( (-1,0)\cup(1,\infty) \)” means increasing on each of the two intervals.
So f is decreasing on \( (-\infty,-1) \) and \( (0,1) \), and increasing on \( (-1,0) \) and \( (1,\infty) \).
Checking the statements
(A) \( f(x) \) is increasing on \( (-\infty,-1) \) — Incorrect. f′(x) < 0 there, so f is decreasing.
(B) \( f(x) \) is decreasing on \( (-\infty,-1) \) — Correct. f′(x) < 0 there.
(C) \( f(x) \) is increasing on \( (-1,0)\cup(1,\infty) \) — Correct. f′(x) > 0 on both intervals.
(D) \( f(x) \) is decreasing on \( (1,\infty) \) — Incorrect. f′(x) > 0 there, so f is increasing.