$\frac 12 \int_{0}^{1} x^4(1-x)^4 dx\leq \int_{0}^{1}\frac{x^4(1-x)^4}{1+x^2}dx \leq \int_{0}^{1} x^4(1-x)^4dx$
This is my homework assignment whereby I have no idea of how should i get started. Could i have some hint on this question??
$\frac 12 \int_{0}^{1} x^4(1-x)^4 dx\leq \int_{0}^{1}\frac{x^4(1-x)^4}{1+x^2}dx \leq \int_{0}^{1} x^4(1-x)^4dx$
This is my homework assignment whereby I have no idea of how should i get started. Could i have some hint on this question??
(**) I hope you know that $\,f(x)\geq 0\,\,\,\,\forall\,x\in [a,b]\Longrightarrow \int_a^bf(x)\,dx\geq 0\,$... If you don't then proving this by means of Riemann sums is really easy.
Well, taking now $\,[0,1]\,$, we have that
$\frac{1}{2}\,x^4(1-x)^4\leq \frac{x^4(1-x)^4}{1+x^2}\leq x^4(1-x)^4$
and now apply (**) to each inequality above
Hint: For $x\in[0,1]$ $\frac12\le\frac1{1+x^2}\le1$ Multiply by $x^4(1-x)^4$ and integrate over $[0,1]$.