Maths Olympiad Prep

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Algebra Difficulty 5.9 AIME, harder Prove it

15. For any real number xx, prove:
5x4+x2+2>5x5 x^{4}+x^{2}+2>5 x

Solution

15. 5x4+x2+2−5x=5(x2−13)2+133(x−1526)2+1468>05 x^{4}+x^{2}+2-5 x=5\left(x^{2}-\frac{1}{3}\right)^{2}+\frac{13}{3}\left(x-\frac{15}{26}\right)^{2}+\frac{1}{468}>0.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.