quasi concave and quasi convex

[N]

determine whether the following is quasiconcave or quasiconvex or both or neither
x^3-x
and
x^3+x

[N]

someone plz solve it

[Amit Goyal]

Since f(x) = (x^3)+x is a monotonically increasing function defined on a real line it is both quasi concave and quasi convex.
g(x) = (x^3) - x is neither quasi concave nor quasi convex. The reason is the following:
Not quasi concave: Consider x = 0 and x' = 1 and t = 1/2,
g(tx + (1-t)x') = g(1/2) = (1/8) - (1/2) = -3/8 < 0 = min{g(x), g(x')}
Not quasi convex: Consider x = 0 and x' = -1 and t = 1/2,
g(tx + (1-t)x') = g(-1/2) = (-1/8) - (-1/2) = 3/8 > 0 = max{g(x),g(x')}