If $f,g,h,\phi$ are polynomials in $x$, and $$p(x)= \left (\int_1^xf(t)h(t)dt\right) \left(\int_1^xg(t)\phi(t)dt\right)-\left(\int_1^xf(t)\phi(t)dt\right) \left(\int_1^xg(t)h(t)dt\right) $$ is divisible by $ (x-1)^\lambda $. Find the maximum real $\lambda$.
- I tried to think that if I find that $(x-1)^m$ divides p’(x) and p’(1)=0, then we can write $\lambda=m+1$ or something like that.
- But differentiating became a major burden.
- p(1)=0, so by factor theorem obviously $\lambda\geq 1$.
- Also, does the lower limit “1” have any real significance? EDIT: I just thought up the fact that any nth derivative which has any of the four involved integrals in it will evaluate to 0 at 1. Is this useful?
If you compute the derivative $$ p'(x) = f(x) h(x) \int_1^x (...)dt+g(x)\phi(x) \int_1^x (...)dt -f(x)\phi(x) \int_1^x (...)dt - g(x)h(x)\int_1^x (...)dt $$
you see that $p'(1)=0$ and so $\lambda \ge 2$. Computing the second derivative, you get $$ p''(1) = 2 f(1)g(1)h(1)\phi(1)-2f(1)\phi(1)g(1)h(1) = 2f(1)g(1)\phi(1)\left(g(1)-h(1)\right)=0 $$
So, we do have $\lambda \ge 3$. Moving on to the third derivative, you'll see that, for general $f,g,\phi,h$ the condition $p'''(1)$ does not hold.