f(x)=2cos^2x + √3 sin2x
= cos2x+1+√3 sin2x
= 2(sin2xcosπ/6+cos2xsinπ/6)+1
=2sin(2x+π/6)+1
f(C)=2
2sin(2C+π/6)+1=2
sin(2C+π/6)=1/2
∵2C+π/6＞π/6
∴2C+π/6=5π/6
∴C=π/3
a/sinA=c/sinC
sinA=asinC/c=4sin(π/3)/5=2√3/5
c^2=a^2+b^2-2abcosC
5^2=4^2+b^2-2*4*b*1/2
b^2-4b=9
(b-2)^2=13
b=2-√13＜0舍去
∴b=2+√13