2011/101) Prove this inequality? 4x^10+x^8+4x^2+1 ≥ 2x^9+4x^5+2x (x is a positive number)

we shall use the AM GM enaquality to prove it

we know (a+b)/2 > = sqrt(ab) or (a+b) > = 2 sqrt(ab)

a = 4x^10, b = x^8 gives 4x^10 + x^ 8 > = 4 x^9
a = 4x^10, b = 1 gives 4x^10 + 1 > = 4 x^5
a = x^8, b = 4x^2 gives x^ 8 +4x^2 > = 4x^5
a = 4x^2, b = 1 gives 4x^2 + 1 > = 4 x

adding we get

8x^10 + 2x^2+ 8x + 2 >=4x^9 + 8x^5 + 2

dividing by 2 we get 4x^10+x^8+4x^2+1 ≥ 2x^9+4x^5+2x