What is the domain of your function?
To show a minimum at x=0, solve f'(x)=0
Let f(x)=(x^(4))(2+sinx^(-1)), if x≠0
...........0, if x=0
Prove that f is differentiable on R (the real numbers).
Prove that f has an absolute minimum at x=0.
Prove that f ' takes on both positive and negative values in every neighborhood of 0.
Pretty lost on this problem, I would really appreciate some help please