most efficient direction to walk down the mountain

Question

Using function $\ f(x,y)=-x^2-y^2$ as the mountain. Currently standing at the point (2,1).

My attempt:

took partial derivative of x and y, then I got

$\ -2x $ and$\ -2y $.

What should I do next? Just simply substitute (2,1) into -2x and -2y?

Answer 1

Indeed. You are looking for the directional derivative at $(2,1)$ and that is how you obtain it. You should be moving in negative gradient direction, so negate the result after you plug it in...

In fact, in optimization algorithms, you now have to solve an additional problem -- not just in which direction, but also how far to go, but this becomes a simple 1d optimization problem we can well deal with, and this is the main principle behind how many multi-dimensional optimization algorithms work.