(adsbygoogle = window.adsbygoogle || []).push({}); The problem statement, all variables and given/known data

Let f: [a,b] -> R be continuous such that f(x) ≥ 0 for all x in [a,b]. Show that

[tex]\lim_{n \to \infty} \left( \int_a^b (f(x))^n \, dx \right)^{1/n} = \max\{f(x) : x \in [a,b]\}.[/tex]

Relevant equations

The fundamental theorem of calculus and its corollaries.

The attempt at a solution

It is easy to show that the limit is less than or equal to the max of f. This doesn't rely on using the fact that f ≥ 0, so somehow this extra piece of info. turns 'less than or equal' into 'equal', but I have failed to determine why. Any tips?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: The Maximum of a Nonnegative Function as an Integral

**Physics Forums | Science Articles, Homework Help, Discussion**