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

1 Answer
Aug 1, 2016

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.