Background
I am looking for some help with the reasoning here for a few things in my game theory book (my math expertise is quite weak - below is from the text):
From Text Book
\begin{eqnarray} min \ u (x, y) = min (4xy-2x-y+3) = min (y(4x-1)-2x+3) \end{eqnarray}
(I cant get stack exchange to accept my other latex so here is a picture)
For each fixed x, this is a linear function in y, and therefore the point at which the minimum is attained is determined by the slope $4x-1$:
if the slope is positive the function is increasing and the minimum is attained at $y=0$;
if the slope is negative this is a decreasing function and the minimum is attained at $y=1$;
if the slope is 0, the function is constant in y and every point is a minimum point. This function of x attains a unique maximum at x= $\frac{1}{4}$, and its value there is 2.5.
What I Don't Get
What calculation is being done to determine that a minimum is attained at $y=0$ if the slope is positive, and that the minimum is attained at $y=1$ if slope is negative?
In the graph picture (linked above), I do not see how the line drawn from the peak max point ($y=2.5, x=1/4$) goes to the point $y=1, x=1$. I understand how the graph starts from the y intercept at 2 (it is the equation y=2x+2), but I don't see how it continues from there with the available data.
Thanks for your time and consideration!