Is the statement, that $\lim_{x \to \infty} {\bigg(\frac {\int_a^b{g^{x}(t)dt}}{b - a}\bigg)}^{\frac {1}{x}} = max(g(t))$, where max(g(t)) is the maximal value, taken by function g(t) on [a, b], true for each real-valued function g(t), that is continuous on [a, b] and takes only positive values on it?
I could neither prove it, nor find any counterexamples to it.
Any help will be appreciated.