Suppose $f$ is a function defined on $-8,8$ given by $f(x)=4x^{\frac{1}{3}}-x^{\frac{4}{3}}-k,$ where $k$ is a positive constant.

a$.$ Show that $f^{\text{'}}(x)=\frac{4-4x}{3x^{\frac{2}{3}}}.$

b$.$ For what values of $x$ is $f(x)$ increasing? Justify your answer.

c$.$ Write an expression for the absolute minimum value of $f$ on $-8,8.$ Show the analysis that leads to your answer.

d$.$ Find all possible values of $k$ such that $f(x)$ has no real zeros. Show the analysis that leads to your answer.