The following theorem was proved by the French mathematician
Rolle, in connection with the problem of approximating roots of polynomials,
but the result was not originally stated in terms of derivatives. In fact,
Rolle was one of the mathematicians who never accepted the new notions of
calculus. This was not such a pigheaded attitude, in view of the fact that
for one hundred years no one could define limits in terms that did not verge
on the mystic, but on the whole history has not been particularly kind to Rolle.
Suppose that $a$ and $b$ are two consecutive roots of a polynomial
function $ f( x),$ but that $a$ and $b$ are not double roots, so that we can
write $ f( x) = (x-a)(x-b) g(x)$ where $ g(a) \neq 0$ and $ g(b)
\neq 0.$
- Prove that $ g(a)$ and $ g(b)$ have the same sign. {(Remember
that a and b are consecutive roots.)}
- Prove that there is some number $x$ with $a Now prove the same fact, even if $a$ and $b$ are multiple roots.
{Hint: If $ f( x) = (x-a)^m(x-b)^n g(x)$ where $ g(a) \neq 0$
and $ g(b) \neq 0,$ consider the polynomial function}
$$ h(x) = \frac{ f'( x) }{ (x-a)^{{m-1}} (x-b)^{{n-1}}}.$$
- A more general result than the latter bears Rolle's name today;
state Rolle's Theorem.
- State the Mean Value Theorem. {(Learn it, Love it, Live it!!)}
Tags: rolle’s-theorem, MVT, IVT, higher-polynomial, proof, history, sketching, multistep, state, a--a, a--g--v, t2, c3
I like to insert a little history whenever possible. My students groan about it, but later they’ll say, “wasn’t Rolle the guy who…” Writing this as a multi-step problem was important for my students because it allowed them to see how the process of moving from a particular result for a polynomial to a generalized result. Of course, I’d repeat (for the nth time at this point) that polynomials are continuous and differentiable on R…I like to remind them of important facts or results we’ve proved when they arise. I think is empowers them and helps them see how things fit together.
– James Epperson 1998-08-09