parmenides51 wrote: Given $x \ge 0$, prove that $$\frac{(x^2 + 1)^6}{2^7}+\frac12 \ge x^5 - x^3 + x$$

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solved also here (as p1)

Let $x \ge 0 .$ Prove that $$\frac{11\sqrt 3+9}{36}\left(x^2 + \frac12\right)^3+\frac18 \ge x^5 - x^3 + x$$Equality holds when $x=\frac{\sqrt 3-1}{2}.$