I am jammed to problem 11 on page 803 here. Earlier on page 797, it claims that the function changes the least when $\langle \nabla f, \bar{e}\rangle=0$ and it changes the most in the direction of the gradient $\nabla f$.
Exapmle
Suppose $f(x,y,z)=\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3}$.
Slowest growth
$\begin{align*} \langle \nabla f, \bar{e}\rangle &= (\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3})\cdot \bar{e}_{x}+ (\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3})\cdot \bar{e}_{y}+ (\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3})\cdot \bar{e}_{z}\\ &= \bar{y}^{2} x +(\bar{x} y^{2}+\bar{z}^{3} y)+(\bar{y}z^{3}) \\&= 0 \end{align*}$
so it is the direction of the slowest growth (according to my book -rule $\langle \nabla f, \bar{e}\rangle =0$, p797)?
Greatest growth direction
$\bar{n}=\max \frac{\nabla f}{\left| f \right|}$ so just I need to find out the inflection points $\nabla^{2} f$ so
$\nabla^{2} f=\begin{pmatrix}\bar{y}^{2}\\ 2\bar{x}\bar{y} \\ 3\bar{y} \bar{z}^{2} \end{pmatrix}=\bar{0},$
I feel I am terribly misunderstanding something.