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

$$a\neq b$$

Prove the next about the solution of this equation.

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

1. All the solution of this equation from $a$ to $b$.

2. All the solution to an equation is different.

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

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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

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 $[a,b]$.

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