Consider the function \( f(x)=x^4-2x^2 \), then which of the following are TRUE?
(A) \( f(x) \) is increasing on \( (-1,0) \)
(B) \( f(x) \) is decreasing on \( (0,1) \)
(C) \( f(x) \) is increasing on \( (-\infty,-1) \)
(D) \( f(x) \) is decreasing on \( (1,\infty) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
The sign of the derivative decides where f increases or decreases.
\( \displaystyle f'(x)=4x^3-4x \)
\( \displaystyle f'(x)=4x(x+1)(x-1) \)
The critical points −1, 0, 1 divide the line into four intervals. Checking the sign of each factor:
\( \displaystyle x\lt -1:\ (-)(-)(-)\ \Rightarrow\ f'(x)\lt 0 \)
\( \displaystyle -1\lt x\lt 0:\ (-)(+)(-)\ \Rightarrow\ f'(x)\gt 0 \)
\( \displaystyle 0\lt x\lt 1:\ (+)(+)(-)\ \Rightarrow\ f'(x)\lt 0 \)
\( \displaystyle x\gt 1:\ (+)(+)(+)\ \Rightarrow\ f'(x)\gt 0 \)
So f is decreasing on \( (-\infty,-1) \), increasing on (−1, 0), decreasing on (0, 1) and increasing on \( (1,\infty) \).
Checking the statements
(A) \( f(x) \) is increasing on \( (-1,0) \) — Correct. f′(x) > 0 there.
(B) \( f(x) \) is decreasing on \( (0,1) \) — Correct. f′(x) < 0 there.
(C) \( f(x) \) is increasing on \( (-\infty,-1) \) — Incorrect. f′(x) < 0 there, so f is decreasing.
(D) \( f(x) \) is decreasing on \( (1,\infty) \) — Incorrect. f′(x) > 0 there, so f is increasing.