Let f be any polynomial function. Prove that there is some number y such that |f(y)| <= |f(x)| for all x.?

Question

Try to incorporate the three hard theorems of continuous functions (i.e. Intermediate Value Theorem, Global Roundedness Theorem, and Extreme Value Theorem).

Answer 1

The region of f[x] is R as it is polynomial

Case a)
f[x] has at least one root at R, let that be x0 <=>
f[x0]=0 <=>
|f[x0]|=0 <=> (we have 0<=|a| for every a in R)
|f[x0]|<=|f[x]|, for every x in R and y=x0 PROVEN

Case b)
f[x] does not have any root at R <=>
f[x]'s degree is even
(Proof:
if f[x]'s degree is odd then:
1)Either lim f[x] (x -> +∞)=+∞ and lim f[x] (x->-∞)=-∞
2)Or lim f[x] (x->+∞)=-∞ and lim f[x] (x->-∞)=+∞
In either case f[x] has a root (from GRT - which also is valid for intervals in the form of: (-∞,+∞) where f[-/+∞]=lim f[x] (x->-/+∞)).)
<=>
1)Either lim f[x] (x->-∞)=lim f[x] (x->+∞)=+∞
2)Or lim f[x] (x->-∞)=lim f[x] (x->+∞)=-∞
<=>
There are x1,x2 that in (-∞,x1] f[x] is genuinely decrasing (or increasing respectively) and [x2,+∞) f[x] is genuinly increasing( or decreasing respectively)
<=>
From GRT we have that in the interval [x1,x2] f[x] reaches both it's max (=M) and min(=m) (in that region)
<=>
Subcase b.1)
If lim f[x] (x->-∞)=lim f[x] (x->+∞)=+∞ then f[x]'s topical min(=m) in [x1,x2] is f[x]'s global min(=m) as for x in (-∞,x1]^[x2,+∞) we have f[x]>=f[m] <=>
0<=f[m]<=f[x], for x in R <=>
|f[m]|<=|f[x]| and y=m PROVEN
Subcase b.2)
If lim f[x] (x->-∞)=lim f[x] (x->+∞)=-∞ then f[x]'s topical max(=M) in [x1,x2] is f[x]'s global max(=M) as for x in (-∞,x1]^[x2,+∞) we have f[x]<=f[M] <=>
0>=f[M]>=f[x], for x in R <=> (notice we change the direction)
0<=|f[M]|<=|f[x]| and y=M PROVEN

In either case and subcase it is proven:
Case a)f[x] has at least a root in R
y=x0 where x0 is any root and so |f[y]|<=|f[x]| for every x in R
Case b)f[x] has not roots in R
Subcase b.1)lim f[x] (x->-∞)=lim f[x] (x->+∞)=+∞
topical min(=m) in [x1,x2] is global min and y=m: |f[y]|<=|f[x]| for every x in R
Subcase b.2)lim f[x] (x->-∞)=lim f[x] (x->+∞)=+∞
topical max(=M) in [x1,x2] is global max and y=M: |f[y]|<=|f[x]| for every x in R