y=x/(x^2+1)y=xx2+1
dy/dx=(1-x^2)/(x^2+1)^2dydx=1−x2(x2+1)2 ( using quotient rule )
"For stationary points, let "dy/dx" be "0","For stationary points, let dydx be 0,
(1-x^2)/(x^2+1)^2=01−x2(x2+1)2=0
1-x^2=01−x2=0
(1-x)(1+x)=0(1−x)(1+x)=0
x=+-1x=±1
"To find the second derivative, differentiate "dy/dx","To find the second derivative, differentiate dydx,
(d^2y)/dx^2=(2x^3-6x)/(x^2+1)^3d2ydx2=2x3−6x(x2+1)3 ( using quotient rule )
When x=1x=1,
(d^2y)/dx^2=(2(1)^3-6(1))/((1)^2+1)^3d2ydx2=2(1)3−6(1)((1)2+1)3
color(white)(xx.)=-1/2>0×.=−12>0 ( maximum point )
When x=-1x=−1,
(d^2y)/dx^2=(x(-1)^3-6(-1))/((-1)^2+1)^3d2ydx2=x(−1)3−6(−1)((−1)2+1)3
color(white)(xx.)=1/2<0×.=12<0 ( minimum point )
Check: graph{x/(x^2+1) [-10, 10, -5, 5]}