Prove-that-a-2-tan-k-x-b-2-sin-k-x-gt-2abx-k-for-all-x-0-pi-2-and-positive-integer-k-

Tinku Tara June 4, 2023 Algebra 0 Comments

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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Prove-that-a-2-tan-k-x-b-2-sin-k-x-gt-2abx-k-for-all-x-0-pi-2-and-positive-integer-k-

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