Question

For what values of x is `f(x)= (x-x^3)/(2-x^3)` concave or convex?

Answer 1

f(x) is concave down in (0,`2^(1/3)`) and concave up in (`2^(1/3), oo`)
&
concave up in (0, `-oo`)

Explanation:

f'' (x) would be `-6x((x^4 -4x^3 +4x -4))/(x^3-2)^3`

For concavity, the second derivative test says that for f(x) is concave up for that value of x for which f''(x)>0 and concave down if f''(x)<0.
From the 2nd derivative shown above, it can be ascertained that for x>0 and up to x<`2^(1/3)`, f''(x) would be <0, at x=`2^(1/3)`, f(x) would not exist and for all x>`2^(1/3)` f''(x)>0.

Hence f(x) is concave down in (0,`2^(1/3)`) and concave up in (`2^(1/3), oo`)

Like wise for all x<0, f''(x) would be positive and hence concave up.