Question

Let f(x) be a polynomial such that 2f(x)+f(2-x)=5+x for any real number x. Find the value of f(0)+f(2)

pls help

Answer 1

Given, f(2−x)=f(2+x)..(i)

∴f(x) is symmetric about x=2

f(20−x)=f(x)..(ii)

x→x+10

f(10−x)=f(10+x)

f(x) is symmetric about x=10

x→x+2

f(18−x)=f(x+2)..(ii)

f(2−x)=f(18−x)

x→−x

f(2+x)=f(18+x)

∴x+2→x

∴f(x)=f(x+16)

f(x) has period = 16

f(x)=5,x∈[0,170]

put x=2 in (i) f(0)=f(4)

put x=4 in (ii) f(4)=f(16)

when x∈[0,16]→f(x) has two solution

when x∈[0,160] has 20 solution.

When xin[0,170] has

′

21

′

solution