I have 3 ordered points, $(x_1,y_1),(x_2,y_2),(x_3,y_3),$ with $x_1<x_2<x_3$, of which I calculate the interpolationg quadratic function. In my case, $(x_2,y_2)$ is a local minimum in or maximum in this list (i.e. ($y_2>y_1$ and $y_2>y_3$) or ($y_2<y_1$ and $y_2<y_3$). My question: is the parabola's minimum or maximum guaranteed to be within the support of these points, so that $x_1<x_\text{min}<x_3$?
I thought about just working out the formula for $x_\text{min}$, but surely there is a more easy argument?
Best Answer
I suppose something like this works?
Assume $x_\text{min} < x_1$. Then if $x_1<x_2<x_3$, $y_1,y_2,y_3$ is a monotonic sequence because the parabola is monotonic after $x_\text{min}$. Similarly for $x_\text{min}>x_3$. Hence, $x_1<x_\text{min}<x_3$.