Given 2 positive reals a and b. There exists 2 polynomials F(x)=x2+ax+b and G(x)=x2+bx+a such that all roots of polynomials F(G(x)) and G(F(x)) are real. Show that a and b are greater than 6.
F(G(x)) can only have real roots, if F(x) has real roots and
G(F(x)) can only have real roots, if G(x) has real roots so:
a^2 >= 4b and b^2 >= 4a. Combining:
a^4 >= (4b)^2 = 16b^2 >= 64a so
a^3 >= 64 meaning a >= 4
Posted by FrankM
on 2020-11-21 14:53:08