Getting to the Root of Things

Suppose the degree of $f(x)$ is $n$ . Let $a_1\leq{a_2}\leq...\leq{a_n}$ be the roots of $f(x)$ , listed with their multiplicities. We will construct roots $b_1\leq{b_2}\leq...\leq{b_{n-1}}$ of $f'(x)$ that are "interlaced" with the roots of $f(x)$ in the sense that $a_1\leq{b_1}\leq{a_2}\leq{b_2}\leq...\leq{b_{n-1}}\leq{a_n}$ .

Indeed, if $a_k<a_{k+1}$ , then Rolle's Theorem guarantees the existence of a $b_k$ between $a_k$ and $a_{k+1}$ with $f'(b_k)=0$ .

If $a_k$ has multiplicity 3, say, with $a_k=a_{k+1}=a_{k+2}$ , then $a_k$ will be a root of $f'(x)$ with multiplicity 2, and we will make $b_k=b_{k+1}=a_k$ . This argument easily generalizes to any multiplicity, $r$ .

The degree of $f'(x)$ is $n-1$ and we have found $n-1$ real roots, so that we have in fact found all the roots of $f'(x)$ , and they are all real as claimed.

$f(x)=(x-a_{ 1 })^{ p_{ 1 } }(x-a_{ 2 })^{ p_{ 2 } }...(x-a_{ k })^{ p_{ k } },\quad where\quad a_{ i }\in R\quad and\quad p_{ i }\in N\\ for\quad all\quad i=1..k,\quad and\quad \sum _{ i=1 }^{ k } p_{ i }=n.\\ \\ When\quad there\quad are\quad no\quad roots\quad of\quad multiplicities,\quad then\quad we\quad have,\\ from\quad a\quad rough\quad plot\quad of\quad f(x),\quad between\quad any\quad two\quad roots,\quad either\quad \\ a\quad maxima\quad or\quad a\quad minima\quad which\quad is\quad a\quad root\quad of\quad f'(x).\quad In\quad this\\ case,\quad we\quad have\quad a\quad derivative\quad with\quad n-1\quad roots\quad and\quad f(x)\quad with\quad \\ n-1\quad extreme\quad points\quad and\quad thus\quad all\quad the\quad roots\quad are\quad accounted\\ for.\\ \\ When\quad there\quad is\quad a\quad multiplicity,\quad say\quad a_{ r }\quad gets\quad repeated\quad twice,\\ then\quad you\quad have,\quad in\quad the\quad derivative\quad f'(x),\quad a_{ r }\quad as\quad a\quad root\quad as\quad well.\\ But\quad now,\quad f\quad has\quad n-1\quad distinct\quad roots\quad and\quad therefore\quad the\quad number\\ of\quad extreme\quad points\quad gets\quad reduced\quad to\quad n-2.\quad So,\quad the\quad roots\quad of\quad \\ f'(x)\quad are\quad accounted\quad for\quad again:\quad the\quad extreme\quad points\quad go\quad down\quad \\ by\quad one\quad and\quad another\quad root \quad( of \quad f(x))\quad pops\quad up.\\ \\ This\quad can\quad be\quad done\quad for\quad any\quad root\quad of\quad any\quad degree\quad of\quad multiplicity.\\ Therefore,\quad a\quad polynomial\quad with\quad all\quad real\quad roots\quad has\quad a\quad derivative\quad with\\ all\quad real\quad roots\quad as\quad well.$

The answer is "True". This follows from the Gauss-Lucas theorem . (This may be more firepower than we need for these specific conditions, but it does the job. Perhaps just an application of Rolle's theorem would be sufficient.)