minimum-value-of-f-x-256-sin-2-x-324-cosec-2-x-x-R-

Question Number 159641 by tounghoungko last updated on 19/Nov/21

Commented by gsk2684 last updated on 19/Nov/21

minimum value of f(x)=256 sin^2 (x)+324 cosec^2 (x) ∀x∈ R f(x)=256 sin^2 (x)+324 cosec^2 (x) f(x)=(16 sin x)^2 +(18 cosec x)^2 f(x)=(16 sin x−18 cosec x)^2 +2(16 sin x)(18cosec x) f(x)=(16 sin x−18 cosec x)^2 +576 min f(x) is 576 or f(x)=256 sin^2 (x)+324 cosec^2 (x) ≥2(√(256sin^2 x . 324cosec^2 x)) =2(16)(18) =576 min f(x) is 576

$${minimum}\:{value}\:{of}\:{f}\left({x}\right)=\mathrm{256}\:\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)+\mathrm{324}\:\mathrm{cosec}\:^{\mathrm{2}} \left({x}\right) \\ $$$$\:\forall{x}\in\:\mathbb{R}\: \\ $$$${f}\left({x}\right)=\mathrm{256}\:\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)+\mathrm{324}\:\mathrm{cosec}\:^{\mathrm{2}} \left({x}\right) \\ $$$${f}\left({x}\right)=\left(\mathrm{16}\:\mathrm{sin}\:{x}\right)^{\mathrm{2}} +\left(\mathrm{18}\:\mathrm{cosec}\:{x}\right)^{\mathrm{2}} \\ $$$${f}\left({x}\right)=\left(\mathrm{16}\:\mathrm{sin}\:{x}−\mathrm{18}\:\mathrm{cosec}\:{x}\right)^{\mathrm{2}} +\mathrm{2}\left(\mathrm{16}\:\mathrm{sin}\:{x}\right)\left(\mathrm{18cosec}\:{x}\right) \\ $$$${f}\left({x}\right)=\left(\mathrm{16}\:\mathrm{sin}\:{x}−\mathrm{18}\:\mathrm{cosec}\:{x}\right)^{\mathrm{2}} +\mathrm{576} \\ $$$${min}\:{f}\left({x}\right)\:{is}\:\mathrm{576} \\ $$$$ \\ $$$${or} \\ $$$${f}\left({x}\right)=\mathrm{256}\:\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)+\mathrm{324}\:\mathrm{cosec}\:^{\mathrm{2}} \left({x}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{2}\sqrt{\mathrm{256sin}\:^{\mathrm{2}} {x}\:.\:\mathrm{324cosec}\:^{\mathrm{2}} {x}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\left(\mathrm{16}\right)\left(\mathrm{18}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{576} \\ $$$${min}\:{f}\left({x}\right)\:{is}\:\mathrm{576} \\ $$

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minimum-value-of-f-x-256-sin-2-x-324-cosec-2-x-x-R-

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