Showing a $g\in L^{q}([0,1])$ and proving an inequality for its $q$-norm

I'm working through some problems dealing with $L^p$ spaces and came across this specific exercise:

Let $1<p, q<\infty$ be such that $\frac{1}{p}+\frac{1}{q}=1$. For any $f \in L^{p}([0,1])$, set$$g(x)=\int_{0}^{x} f(t) dt.$$Prove that $g \in L^{q}([0,1])$ and $\|g\|_{q} \leq 2^{-1 / q}\|f\|_{p}$.

I'd like to apply Hölder's Inequality with ($f(t) \cdot 1$) to proceed, but the details are escaping me. Would hugely appreciate a hint on how to efficiently navigate the relevant inequalities!