  Let $\{Q^{(\alpha)}_{n,\lambda}\}_{n\geq 0}$ be the sequence of monic
orthogonal polynomials with respect the Gegenbauer-Sobolev inner product
$$\langle
f,g\rangle_{S}:=\int_{-1}^{1}f(x)g(x)(1-x^{2})^{\alpha-\frac{1}{2}}dx+\lambda
\int_{-1}^{1}f'(x)g'(x)(1-x^{2})^{\alpha-\frac{1}{2}} dx,$$ where
$\alpha>-\frac{1}{2}$ and $\lambda\geq 0$. In this paper we use a recent result
due to B.D. Bojanov and N.
  Naidenov \cite{BN2010}, in order to study the maximization of a local
extremum of the $k$th derivative $\frac{d^k}{dx^k}Q^{(\alpha)}_{n,\lambda}$ in
$[-M_{n,\lambda}, M_{n,\lambda}]$, where
  $M_{n,\lambda}$ is a suitable value such that all zeros of the polynomial
$Q^{(\alpha)}_{n,\lambda}$ are contained in $[-M_{n,\lambda}, M_{n,\lambda}]$
and the function $\left|Q^{(\alpha)}_{n,\lambda}\right|$ attains its maximal
value at the end-points of such interval. Also, some illustrative numerical
examples are presented.
