It is given that
f(x) = [x3 + 1/x3]
By differentiating w.r.t. x
f’(x) = 3x2 – 3x-4
By taking 3 as common
f’(x) = 3[x2 – 1/x4]
Taking LCM
f’(x) = 3[(x6 – 1)/x4] = 3[(x2)3 – 1]/x4
On further calculation
f’(x) = [3(x2 – 1) (x4 + x2 + 1)]/x4
So we get
f’(x) = [3(x – 1) (x + 1) (x4 + x2 + 1)]/x4 < 0 for x ∈ (- 1, 1).
Therefore, f(x) is decreasing function on ]-1, 1[.