partial derivatives?

Question

can you devise a relationship between the signs of
partial of f with respect to x
partial of f with respect to y
partial of g with respect to x
partial of f with respect to y

given two multivariable functions f and g and

grad(f) * grad (g)=0??

* means dot product

Answer 1

Start by expanding the dot product:

f,x *g,x +f,y*g,y +f,z*g,z = 0 where the ",x" means partial derivative with respect to x and similar of teh ",y" and ",z".

Let's assume that both f and g are indpendent of z since your question ignore tis partial derivative. Then the last term is zero and we have:

f,x*g,x +f,y*g,y = 0 ---> f,x*g,x = - f,y*g,y

Now both f,x and g,x can have negative signs or they can both have positive signs, and the above relationship will hold true. Both f,y and g,y can have negative signs or they can both have positive signs, and the above relationship will hold true.

Now if either f,x or g,x is negative, then either f,y or g,y must be negative for teh above relationship to be true.

Answer 2

Write it out:

f_x g_x +f_y g_y=0.

Thus, the product of f_x and g_x has the opposite sign as that of f_y g_y.