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Question Number 158534 by HongKing last updated on 05/Nov/21

Prove that   a^2 tan^k x+b^2 sin^k x > 2abx^k   for all  x∈(0 ; (π/2)) and positive integer k

$$\mathrm{Prove}\:\mathrm{that}\:\:\:\mathrm{a}^{\mathrm{2}} \mathrm{tan}^{\boldsymbol{\mathrm{k}}} \mathrm{x}+\mathrm{b}^{\mathrm{2}} \mathrm{sin}^{\boldsymbol{\mathrm{k}}} \mathrm{x}\:>\:\mathrm{2abx}^{\boldsymbol{\mathrm{k}}} \\ $$ $$\mathrm{for}\:\mathrm{all}\:\:\mathrm{x}\in\left(\mathrm{0}\:;\:\frac{\pi}{\mathrm{2}}\right)\:\mathrm{and}\:\mathrm{positive}\:\mathrm{integer}\:\boldsymbol{\mathrm{k}} \\ $$ $$ \\ $$

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