Let \(K\) be a compact subset of the interval \([a,b]\) contained at least \(n+1\) points and let \(C(K)\) be the Banach space (with respect to the sup-norm) of all continuous real-valued functions defined on \(K\). Let also \(\{\varphi_1, \dots, \varphi_n\}\) be a linearly independent system of continuous functions on \([a,b]\) having derivatives of order \(s(s\in\mathbb{N})\) and let \(\Phi_n= \text{span} \{\varphi_1, \dots, \varphi_n\}\). For \(h_s\), \(u_s: [a,b]\to \overline\mathbb{R}\), \(-\infty\leq h_s(x)\leq u_s(x)\leq +\infty\) put \(K_s:= \{q\in\Phi_n: h_s(x)\leq q^{(s)} (x)\leq u_s(x)\), \(x\in [a,b]\}\). In the first theorem of this paper the author gives a necessary and sufficient condition in order that \(p\in K_s\) be a best approximation element for \(f\in C(K) \setminus K_s\). If \(s_1,s_2, \dots, s_k\in\mathbb{N}\) and \(K_S= \cap^k_{i=1} K_{s_i}\), in Theorem 2 one obtains a necessary and sufficient condition in order that \(p\in K_S\) be an element of best approximation for \(f\in C(K) \setminus K_S\). The above theorems contain as particular cases the approximation with Hermite-Birkhoff interpolatory side conditions, multiple comonotone approximation, etc.