Question

Prove the following inequality when x is positive: 4x^10+x^8+4x^2+1≥2x^9+4x^5+2x

Answer 1

Take the log of both sides

Answer 2

log base x
Log(4)+10+8+Log(4)+2+1 >= Log(2) + 9 + Log(4) + 5 + Log(2) + 1
Use properties of logs to combine and subtract from both sides
Log(16) + 21 >= Log(16) + 15
21 >= 15

Answer 3

I am not sure of you got to the first line. Won't it remain as log(base x)(4x^10+x^8+4x^2+1)≥log(base x)(2x^9+4x^5+2x) ? How could you break it up like that? Please explain. Thanks

Answer 4

You're right I did it wrong, let me look for another way

Answer 5

I couldn't find any better proof but, divide both sides by the right hand side to get
(4x^10+x^8+4x^2+1)/(2x^9+4x^5+2x) > 1. Find the minimum of the left hand side by taking the derivative and solving for 0. (Ugly). You get x=.930187 for a minimum. (only positive minimum) Plugging that into the left hand side you get 1.224, which is always > 1.