Solutions of $n$th derivative of $(x-a)^n(x-b)^n=0$

Question

$$ a\neq b$$

Prove the next about the solution of this equation.

$$ \frac{d^n}{dx^n}(x-a)^n(x-b)^n=0 $$

All the solution of this equation from $a$ to $b$.
All the solution to an equation is different.

Probably, it is the problem to which Legendre polynomial is related.

What does (1.) mean? I just can't make sense of that. The same with (2.)...??? — DonAntonio, Oct 22 '12 at 12:47
If I should guess, I'd say: (1) All $n$ solutions are elements of $[a,b]$. (2) All $n$ solutions are different. — martini, Oct 22 '12 at 14:07

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

Let's make the change of variables $$ y=\frac{2}{b-a}\left(x-\frac{a+b}{2}\right) $$ After substitution and simplifications, we get $$ \frac{d^n}{dy^n}(y^2-1)^n=0 $$ which is equivalent to $P_n(y)=0$, where $P_n$ is a Legendre polynomial. For Legendre polynomials, it is known that the roots are all different and are within $[-1,1]$; for details, see the discussion in Roots of Legendre Polynomial. Hence, the roots of the original equation are also different and are within the interval $[a,b]$.

Solutions of $n$th derivative of $(x-a)^n(x-b)^n=0$

1 Answers1