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Another way toTo see that $f_{1}$, $f_{2}$, $\ldots$, $f_{k}$ must be scalar multiples of $f_{k+1}$ is to observe that $f_{k+1}^{2}$ must have a local minimum at, plug in the root of each$f_{i}$$f_{k+1}$ into both sides of the equation.
Another way to see that $f_{1}$, $f_{2}$, $\ldots$, $f_{k}$ must be scalar multiples of $f_{k+1}$ is to observe that $f_{k+1}^{2}$ must have a local minimum at the root of each$f_{i}$.
To see that $f_{1}$, $f_{2}$, $\ldots$, $f_{k}$ must be scalar multiples of $f_{k+1}$, plug in the root of $f_{k+1}$ into both sides of the equation.
Another way to see that $f_{1}$, $f_{2}$, $\ldots$, $f_{k}$ must be scalar multiples of $f_{k+1}$ is to observe that $f_{k+1}^{2}$ must have a local minimum at the root of each $f_{i}$.
