If λ and M be real numbers such that X³-λX²+mX-6=0 has its roots real and positive, then minimum value of M is

X³-λX²+mX-6=0

a+b + c  = λ

ab + bc + ca = m

abc = 6

applying A.M.   G.M. inequality,

(ab + bc + ca )/3 >=  ( ab.bc.ca) ^1/3

                        >  =  ( abc) ^2/3

                        > =    6^2/3

(ab + bc + ca )   >= 3 *  (6) ^2/3

 so its minimum value will be = 3 *  (6) ^2/3