I'm having trouble to articulate how the actual graph looks like when z = 0, meaning in the middle of the level curve where intersection occurs.
What shape would the graph look like and what curvature is it supposed to have?
What does it imply when two lines cross each other in a level curve
graphing-functionsmultivariable-calculus
Related Solutions
You could imagine that the medicine gets absorbed by the digestive system at a fast rate $\alpha$ and then consumed by the body at a slower rate $\beta$. Then you have the following system of ordinary differential equations, $$\begin{align} x' &= -\alpha x, \\ y' &= \alpha x - \beta y, \end{align}$$ where $x$ and $y$ are the amounts of unabsorbed medicine in the stomach and absorbed but unconsumed medicine in the bloodstream respectively. For initial conditions $x(0) = q$ and $y(0) = 0$, the solution is simply $$\begin{align} x(t) &= e^{-\alpha t}q, \\ y(t) &= -\frac{\alpha}{\alpha-\beta}e^{-(\alpha+\beta)t}\left(e^{\beta t}-e^{\alpha t}\right)q. \end{align}$$ For $q = 1$, $\alpha = 1$, $\beta = 0.1$, the curve looks like this:
$xy=3$ will be a rectangular hyperbola, just like other curves of the form $xy=\text{constant}$.
Best Answer
This is a saddle point, like a Pringle. If you move in certain direction away from the origin (positive or negative $x$-directions) the function will increase, while if you move away from the origin in other directions (positive or negative $y$-directions) the function will decrease.