Show that 3x^4+1 is O(x^4/2) and x^4/2 is O(3x^4+1)...

My teacher gave us this hint: (hint: prove x^4 is not O(x^3) by contradiction)

Please help me if possible. This is Discrete Mathematics and I am extremely confused on what to do..

Note that for all x > 2
3x^4 + 1
= 6(x^4/2) + 1
< 6(x^4/2) + x^4/2
= 7 * (x^4/2).

Since 3x^4 + 1 < 7 * (x^4/2) for all x > 2, we conclude that 3x^4 + 1 is O(x^4/2).
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2) Similarly,
3x^4 + 1
= 6(x^4/2) + 1
> 6(x^4/2)

So, x^4/2 < (1/6) (3x^4 + 1) for all x > 0.
Hence, x^4/2 is O(3x^4 + 1).

I hope this helps!