The direction you'll walk to descend fastest is obtained as follows:
[tex]u= \frac{-\nabla z(5,\ 4)}{||\nabla f(5,\ 4)||} \\ \\ = \frac{-\ \textless \ -4x,\ -6y\ \textgreater \ |_{(5,\ 4)}}{||\ \textless \ -4x,\ -6x\ \textgreater \ ||_{(5,\ 4)}} \\ \\ = \frac{\ \textless \ 20,\ 24\ \textgreater \ }{\sqrt{(-20)^2+(-24)^2}} = \frac{\ \textless \ 20,\ 24\ \textgreater \ }{\sqrt{400+576}} \\ \\ = \frac{\ \textless \ 20,\ 24\ \textgreater \ }{\sqrt{976}} = \frac{\ \textless \ 20,\ 24\ \textgreater \ }{4\sqrt{61}} = \frac{\ \textless \ 5,\ 6\ \textgreater \ }{\sqrt{61}} [/tex]
