For the function \( f(x)=\displaystyle\frac{1}{x^{2}+2x+2} \)
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 decreasing on \( (-1,\infty) \)
D. maximum value of \( f(x) \) is 1
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, C and D only
\( \displaystyle f'(x)=-\frac{2x+2}{(x^{2}+2x+2)^{2}} \)
\( \displaystyle =-\frac{2(x+1)}{(x^{2}+2x+2)^{2}} \)
The denominator is always positive, so the sign of f′(x) is the sign of \( -(x+1) \):
\( x\lt -1:\ f'(x)\gt 0 \)
\( x\gt -1:\ f'(x)\lt 0 \)
So f increases up to x = −1 and decreases after it, and its maximum is at x = −1:
\( \displaystyle f(-1)=\frac{1}{1-2+2} \)
\( \displaystyle =1 \)
So f is increasing on \( (-\infty,-1) \), decreasing on \( (-1,\infty) \), and its maximum value is 1.
Checking the statements
(A) \( f(x) \) is increasing on \( (-\infty,-1) \) — Correct. f′(x) > 0 for x < −1.
(B) \( f(x) \) is decreasing on \( (-\infty,-1) \) — Incorrect. f′(x) > 0 there, so f is increasing.
(C) \( f(x) \) is decreasing on \( (-1,\infty) \) — Correct. f′(x) < 0 for x > −1.
(D) maximum value of \( f(x) \) is 1 — Correct. The maximum is f(−1) = 1.
Shortcut: Write the denominator as \( (x+1)^{2}+1 \). It decreases on \( (-\infty,-1) \), increases on \( (-1,\infty) \) and is least (= 1) at x = −1. Its reciprocal f does the opposite, so A, C and D follow at once.