Consider the Cauchy bound for the roots of a complex polynomial $f(x) = a_n x^n + a_{n-1}x^{n-1}\cdots + a_0$, which states that the moduli of all zeros of $f$ are less or equal to the unique positive root of the Cauchy polynomial $$g(x) = |a_n|x^n - |a_{n-1}|x^{n-1} - \cdots - |a_0|.$$ The fact that $g(x)$ has a unique positive root follows immediately from Descartes rule of signs.
According to my professor this bound is supposed to be the sharpest possible bound for the moduli of the zeros of a polynomial, if only the moduli of the coefficients are known. He refused to give an explanation because, in his words, it would be obvious to see for myself, but it isn't.
I am not looking for a rigorous proof, but can anyone explain why this is the case?