Question

Find the maximum rate of change of f(x,y,z=x y/z at the point (2, -3, 5 and the direction in which it occurs. maximum rate of change:

Answer 1

`f(x,y,z)=x+\frac yz`

`\implies \\abla f(x,y,z)=\left(1,\frac1z,-\frac y{z^2}\right)`

`\implies \\abla f(2,-3,5)=\left(1,\frac15,\frac3{25}\right)`

The maximum rate of change is then


`\|\\abla f(2,-3,5)\|=(√(659))/(25)`

which occurs in the direction of
`\\abla f(2,-3,5)` above.