On what interval is the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ increasing?
A) $[ - 1,1 ]$

B) $[ - 1,2 ]$
C) $[ - 2,1 ]$
D) $[ - 2,2 ]$
E) $( - \infty , 1 ]$
F) $( - \infty , 2 )$

G) $[ - 2 , \infty )$

H) $( - \infty , \infty )$

Question

On what interval is the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ increasing?
A) $[ - 1,1 ]$

B) $[ - 1,2 ]$ 
C) $[ - 2,1 ]$ 
D) $[ - 2,2 ]$ 
E) $( - \infty , 1 ]$
F) $( - \infty , 2 )$

G) $[ - 2 , \infty )$

H) $( - \infty , \infty )$

Quizplus · Accepted Answer

To determine where the function is increasing, we find its derivative and analyze its sign. The derivative of $f(x) = \frac{x}{x^2 + 1}$ is $f'(x) = \frac{(1)(x^2 + 1) - (x)(2x)}{(x^2 + 1)^2} = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}$. The derivative is positive (indicating an increasing function) when $1 - x^2 > 0$, which simplifies to $x^2 < 1$, or $-1 < x < 1$. Therefore, the function is increasing on the interval $[-1, 1]$.

On What Interval Is the Function f(x)=xx2+1f ( x ) = \frac { x } { x ^ { 2 } + 1 }f(x)=x2+1x​

On What Interval Is the Function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$