Suppose y(x) is strictly increasing and y'''(x)=0.
Since it's an increasing function, for dx > 0 I know:
dy = y'dx + .5y''dx^2 > 0.
Dividing by dx:
dy/dx = y' + .5y''dx > 0.
But what if I instead take partial derivative of dy wrt dx? That is:
\partial (dy)/\partial (dx) = y' + y''dx.
I now have an extra .5y''dx. Do I still know this is positive? Since y(x) is increasing, it seems that dy grows with dx, so intuitively it seems so. Also, I know for small enough dx, the sign of (y' + y''dx) is the same as the sign of y', which is positive. But on the other hand y'' can be any sign and dx might be large. So it seems not.
Which is right? Thanks in advance for your help.