Can you prove the inequality challenge?

[anemone]

Let $x\ge \dfrac{1}{2}$ be a real number and $n$ a positive integer. Prove that $x^{2n}\ge (x-1)^{2n}+(2x-1)^n$.

 

[Opalg]

Let $x\ge \dfrac{1}{2}$ be a real number and $n$ a positive integer. Prove that $x^{2n}\ge (x-1)^{2n}+(2x-1)^n$.

[sp]If $a$ and $b$ are positive then $(a+b)^n \geqslant a^n+b^n$ (because the binomial expansion of the left side consists of the two terms on the right side, together with other terms which are all positive). Put $a=2x-1$ and $b = (x-1)^2$. Then $a+b = x^2$ and the inequality becomes $x^{2n} \geqslant (2x-1)^n + (x-1)^{2n}.$ [/sp]

 

[anemone]

Well done, Opalg(Yes) and thanks for participating!:)