Consider continuous functions $f,g:(0,1)\rightarrow(0,\infty)$ such that $$\int_0^1f(x)dx=\int_0^1g(x)dx$$ and there is a subinterval $(a,b)\subseteq[0,1]$ such that $f(x)\neq g(x)$ for all $x\in (a,b)$.
Define the following integral for integers $n\ge0$. $$I_n=\int_0^1\frac{[f(x)]^{n+1}}{[g(x)]^n}dx$$
Which of the following is true?
- $I_1>I_0$
- $I_1<I_0$
- $I_1=I_0$
- None of the above
My attempt
I substituted $g(x)=1$ and $f(x)=2x$ and got check the value of integral and got option 1. I want to know how to do this properly. Hints please.
Hint: $$\eqalign{I_1 - I_0 &= \int_0^1 \left( \frac{f(x)^2}{g(x)} - f(x) \right)\; dx - \int_0^1 (f(x) - g(x))\; dx\cr &= \int_0^1 \frac{(f(x) - g(x))^2}{g(x)} \; dx}$$