Maths Olympiad Prep

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, 2012

Algebra Difficulty 4.5 AIME Prove it Saudi Arabia

Prove that for every real number xx the following inequality holds:
x6+x4−x3−x+34>0. x^6 + x^4 - x^3 - x + \frac{3}{4} > 0.

Solution

The inequality is equivalent to
x6−x3+14+x4−x2+14+x2−x+14>0, x^6 - x^3 + \frac{1}{4} + x^4 - x^2 + \frac{1}{4} + x^2 - x + \frac{1}{4} > 0,
which in turn is equivalent to
(x3−12)2+(x2−12)2+(x−12)2>0. \left(x^3 - \frac{1}{2}\right)^2 + \left(x^2 - \frac{1}{2}\right)^2 + \left(x - \frac{1}{2}\right)^2 > 0.

Since 12≠12\frac{1}{\sqrt{2}} \neq \frac{1}{2}, equality cannot occur.

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