Show that if $\displaystyle f $ is a continuous real-valued function on the interval $\displaystyle [a,b] $ and $\displaystyle f(x) \geq 0 \; \forall x \in [a,b] $, then

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