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## Homework Statement

The evaluated partial derivative of f(x,y) with respect to x is -16 and 6 with respect to y at some point (x0,y0). What is the vector specifying the direction of maximum increase of f?

## Homework Equations

The direction of maximum increase of f is given by [itex]\nabla[/itex]f(x,y). The maximum value of D[itex]_{u}[/itex]f(x,y) is ||grad f(x,y)||.

## The Attempt at a Solution

I know the answer is -16i + 6j. But I just don't know why. I understand the geometric argument based on the dot product given in my textbook for why the gradient gives the direction of maximum increase. That's fine. But is there a more intuitive way to look at it in terms of partial derivatives so that the result can be easily extended to higher dimensions?

The main part that's confusing me is that the rate of change of f wrt x is negative (-16) and yet we want to move partly in the x direction. It seems like we would only want to move in the y direction since the rate of change of f wrt y is positive.

Any help would be much appreciated.

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