For $n\ge14$, I need to show that $$(n-4)(n-2)\sqrt{n^2-10n+33}+n^2<(n-3)\sqrt{(n-2)^2+(n-3)^4}.$$ How to show it.
2026-03-25 04:41:15.1774413675
To prove an inequality of degree 3 polynomilals
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It's enough to prove that $$(n-3)^3-n^2>(n-4)(n-2)\sqrt{n^2-10n+33},$$ which after squaring of both sides gives $$(2n-3)(n^4-24n^3+173n^2-476n+461)>0,$$ which is true because $$n^4-24n^3+173n^2-476n+461=(n^2-12n-28)^2+85n(n-14)+42n-323>0.$$