  We consider polynomials of degree $d$ with only real roots and a fixed value
of discriminant, and study the problem of minimizing the absolute value of
polynomials at a fixed point off the real line. There are two explicit families
of polynomials that turn out to be extremal in terms of this problem. The first
family has a particularly simple expression as a linear combination of $d$-th
powers of two linear functions. Moreover, if the value of the discriminant is
not too small, then the roots of the extremal polynomial and the smallest
absolute value in question can be found explicitly. The second family is
related to generalized Jacobi (or Gegenbauer) polynomials, which helps us to
find the associated discriminants. We also investigate the dual problem of
maximizing the value of discriminant, while keeping the absolute value of
polynomials at a point away from the real line fixed. Our results are then
applied to problems on the largest disks contained in lemniscates, and to the
minimum energy problems for discrete charges on the real line.
