Rational functions of best approximation in integral metrics

We prove that any rational function of degree $\le n$ with real coefficients best approximating a function $f\in L_q(a,b)$, $1<q<\infty$, in the metric of $L_q(a,b)$ is of degree exactly $n$. We find the signs of the polynomials best approximating in the metric of $L_q([a,b])$, $1\le q\le\infty$ for some classes of differentiate functions.