Problem:
$f(x)$ and $g(x)$ are two functions with respect to $x$, and they are continuous and differential. Now, I need to plot a curve composed of $(f(x),g(x))$, e.g., the curve is the connection of these points $\{(f(x0),g(x0)), (f(x1),g(x1)), (f(x2),g(x2))...\}$
Then, how to justify the shape of this curve? Is it convex or concave? Thanks.
You have a parametric curve,
$$ r(t) = (f(t) , g(t)$$ which is a curve on the plane.
In order to discuss the concavity of this curve with respect to the x-axis, you need to find $$\frac {d^2 y}{dx^2}$$
Note that $$\frac {dy}{dx} = \frac {\frac {dy}{dt}}{ \frac {dx}{dt}} = \frac {g'}{f'} $$ using Quotient rule we get, $$\frac {d^2 y}{dx^2}=\frac {f'g'' -f''g'}{f'^2} $$