Problem

A function $\mathrm{f}$ with domain $x>0$ is such that $\mathrm{f}'(x)=8(2x-3)^{\frac{1}{3}}-10x^{\frac{2}{3}}$. It is given that the curve with equation $y=\mathrm{f}(x)$ passes through the point $(1,0)$.

(a) Find the equation of the normal to the curve at the point $(1,0)$.

(b) Find $\mathrm{f}(x)$.

It is given that the equation $\mathrm{f}'(x)=0$ can be expressed in the form

$$125x^2-128x+192=0.$$

(c) Determine, making your reasoning clear, whether $\mathrm{f}$ is an increasing function, a decreasing function or neither.