Question Number 33167 by abdo imad last updated on 11/Apr/18
![f is a continue and positive function on [a,b] with a<b let m =max_(x∈[a,b]) f(x) prove that lim_(n→∞) ( (1/(b−a)) ∫_a ^b f^n (x)dx)^(1/n)](https://www.tinkutara.com/question/Q33167.png)
$${f}\:{is}\:{a}\:{continue}\:{and}\:{positive}\:{function}\:{on}\:\left[{a},{b}\right]\:{with}\:{a}<{b} \\ $$$${let}\:{m}\:={max}_{{x}\in\left[{a},{b}\right]} \:{f}\left({x}\right)\:{prove}\:{that} \\ $$$${lim}_{{n}\rightarrow\infty} \:\:\left(\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:{f}^{{n}} \left({x}\right){dx}\right)^{\frac{\mathrm{1}}{{n}}} \\ $$
Commented by abdo imad last updated on 12/Apr/18
$${prove}\:{that}\:{m}={lim}_{{n}\rightarrow\infty} \left(\:\frac{\mathrm{1}}{{b}−{a}}\int_{{a}} ^{{b}} \:{f}^{{n}} \left({x}\right){dx}\right)^{\frac{\mathrm{1}}{{n}}} \:. \\ $$