If roots of the equation
x4−8x3+bx2+cx+16=0
are positive, then
b=24
If x1,x2,x3,x4 are four roots of (1), then
x1+x2+x3+x4=8and x1 x2 x3 x4=16
As A.M. ≥ G.M., we get
14(x1+x2+x3+x4)≥(x1 x2 x3 x4)1/4=2i.e. A.M.=G.M.Thus, x1=x2=x3=x4=2∴x4−8x3+bx2+cx+16=(x−2)4⇒b=24 and c=−32