# Problem 43

• Dec 4th 2007, 08:26 PM
ThePerfectHacker
Problem 43
1)Let $\displaystyle a_1,a_2,...,a_n$ be real (or complex) distinct numbers where $\displaystyle n\geq 2$. Define $\displaystyle P_k = \prod_{j\not = k} (a_k - a_j)$. Can it be that $\displaystyle P_1=P_2=...=P_n$?
(For example, let $\displaystyle a_1=1,a_2=2,a_3=3$ then $\displaystyle P_1 = (1-2)(1-3)=2$, $\displaystyle P_2 = (2-1)(2-3)=-1$, and $\displaystyle P_3=(3-1)(3-2)=2$ but all three are not the same in this case).

The next problem is for the younger kids so please give them a chance.

2)Let $\displaystyle f(x)$ and $\displaystyle g(x)$ be functions which you can differenciate. Note that $\displaystyle [f(x)g(x)]' = f'(x)g(x)+f(x)g'(x)$. Can you find a formula for $\displaystyle [f(x)g(x)]^{(n)}$ where by $\displaystyle ^{(n)}$ means to preform differenciation $\displaystyle n$ times repeatedly.
• Dec 4th 2007, 09:28 PM
DivideBy0
Quote:

Originally Posted by ThePerfectHacker
2)Let $\displaystyle f(x)$ and $\displaystyle g(x)$ be functions which you can differenciate. Note that $\displaystyle [f(x)g(x)]' = f'(x)g(x)+f(x)g'(x)$. Can you find a formula for $\displaystyle [f(x)g(x)]^{(n)}$ where by $\displaystyle ^{(n)}$ means to preform differenciation $\displaystyle n$ times repeatedly.

After a little experimenting:

$\displaystyle [f(x)g(x)]'=f'(x)g(x) + f(x)g'(x)$

$\displaystyle [f(x)g(x)]''=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x)$

$\displaystyle [f(x)g(x)]''' = f'''(x)g(x)+3f''(x)g'(x)+3f'(x)g''(x)+f(x)g'''(x)$

This reminds me of the binomial theorem:

$\displaystyle (x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k} y^{k}$

So perhaps

$\displaystyle [f(x)g(x)]^{(n)} = \sum_{k=0}^n {n\choose k} f^{n-k}(x) g^{k}(x)$

Where $\displaystyle h^n(x)$ denotes the nth derivative of $\displaystyle h(x)$

Nice problem :D

P.S Is there any way to hide text?
• Dec 5th 2007, 06:11 AM
topsquark
Quote:

Originally Posted by DivideBy0
P.S Is there any way to hide text?

Color the text white.

-Dan
• Dec 5th 2007, 08:51 AM
ThePerfectHacker
Quote:

Originally Posted by DivideBy0
Nice problem :D

Good job. This result is called "Leibniz's Rule".
• Dec 11th 2007, 08:06 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
1)Let $\displaystyle a_1,a_2,...,a_n$ be real (or complex) distinct numbers where $\displaystyle n\geq 2$. Define $\displaystyle P_k = \prod_{j\not = k} (a_k - a_j)$. Can it be that $\displaystyle P_1=P_2=...=P_n$?
(For example, let $\displaystyle a_1=1,a_2=2,a_3=3$ then $\displaystyle P_1 = (1-2)(1-3)=2$, $\displaystyle P_2 = (2-1)(2-3)=-1$, and $\displaystyle P_3=(3-1)(3-2)=2$ but all three are not the same in this case).

Since I know the way your mind works, there is something wrong with what I am about to say... :)

Consider the case n = 3. Then the question is can
$\displaystyle a_1a_2 = a_1a_3 = a_2a_3$
where all three are distinct?

Obviously not since $\displaystyle a_1a_2 = a_1a_3 \implies a_2 = a_3$.

Now, there's got to be something screwy here because I can generalize this argument to larger n and come up with similar results. But I can't believe it would be this easy... (Sweating)

-Dan
• Dec 11th 2007, 08:37 AM
ThePerfectHacker
I came up with problem #1 accidently when I playing around with polynomials. Here is my original solution, however it seems to me that this problem is easy even if approached directly.

Proof:
Let $\displaystyle a_1,...,a_n$ be real (or complex numbers) and define $\displaystyle f(x) = (x-a_1)...(x-a_n)$. The key step is to note that $\displaystyle f'(x_k) = P_k$ by using the general product rule for derivatives. Now, $\displaystyle \deg f(x) = n\geq 2$ thus, $\displaystyle \deg f'(x) = n-1\geq 1$, this means that $\displaystyle f'(x)$ cannot attain the same values at $\displaystyle a_1,a_2,...,a_n$ (meaning $\displaystyle f(a_1)=f(a_2)=...=f(a_n)$) because otherwise the situation is that a degree $\displaystyle n-1$ polynomial attains the same value $\displaystyle n$ times for $\displaystyle n$ distinct numbers. Which is impossible. Thus, $\displaystyle P_1,P_2,...,P_n$ cannot all be the same.