What can we say about the function ux y at 00 if delta u001S

What can we say about the function u(x, y) at (0,0) if delta u(0,0)=-1?

Solution

The derivative at the point (0,0) for u(x,y) is negative implies that the function is decreasing at the point (0,0), i.e. the value of the function at (0+h,0+h) will be higher than the value at (0,0) and the value of the function at (0-h,0-h) will be lower than 0

The derivative exists also implies that the function is continuous and will be going to decrease in the region around (0,0)