(Solved): Please do 1 and 3. If you don't know the answer, please don't try. Thanks 1. Let f:[0,1](0, ...

Please do 1 and 3.
If you don't know the answer, please don't try.

Thanks

1. Let $$f:[0,1]?(0,?)$$ be a continuous function, such that $$L:=\underset{n??}{\lim }{?}_0^1(f(x){)}^{n}dx$$ exists and is finite. Show that $$L?[0,1]$$. Hint: Show first that $$0<f(x)?1$$. 2. Show that if $$f:[0,1]?\mathbf{R}$$ is a continuous function, then $$\underset{n??}{\lim }{\left({?}_0^1?f(x){?}^{n}dx\right)}^{\frac{1}{n}}=\underset{[0,1]}{\max }?f(x)?.$$ 3. Let $$g:(0,?)?\mathbf{R}$$ is a an increasing function, so that $$\underset{x??}{\lim }g(x)=?$$. Also, $$f:(0,?)?\mathbf{R}$$ is a Riemann integrable function, with $${?}_0^{?}?f(x)?dx<?$$. Prove that $$\underset{R??}{\lim }\frac{{?}_0^{R}g(x)f(x)dx}{g(R)}=0$$