The correct minimum is (-1/√3 or -√3/3,-3√3/16) and the correct maximum is (1/√3 or √3/3,3√3/16). This solution is not listed in the options.
Here’s the full solution.
f(x)=x/(x²+1)², df/dx=[(x²+1)²-(x)(2(x²+1))(2x)]/(x²+1)⁴=
(x⁴+2x²+1-4x²(x²+1))/(x²+1)²=0 at critical points.
-3x⁴-2x²+1=0=-(3x²-1)(x²+1), so x²=⅓ and x=√⅓ or -√⅓, which can be written √3/3, -√3/3.
f(0)=0, so we can see which is a min and which is a max:
f(√⅓)=3√3/16 (max), f(-√⅓)=-3√3/16 (min).