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

$$minimumvalueoff\left(x\right)=256\sin {}^2\left(x\right)+324\mathrm{cosec}{}^2\left(x\right)$$$$\mathrm{\forall }x\in \mathbb{R}$$

Commented by gsk2684 last updated on 19/Nov/21

$$minimumvalueoff\left(x\right)=256\sin {}^2\left(x\right)+324\mathrm{cosec}{}^2\left(x\right)$$$$\mathrm{\forall }x\in \mathbb{R}$$$$f\left(x\right)=256\sin {}^2\left(x\right)+324\mathrm{cosec}{}^2\left(x\right)$$$$f\left(x\right)={\left(16\sin x\right)}^2+{\left(18\mathrm{cosec}x\right)}^2$$$$f\left(x\right)={(16\sin x-18\mathrm{cosec}x)}^2+2\left(16\sin x\right)\left(18\mathrm{cosec}x\right)$$$$f\left(x\right)={(16\sin x-18\mathrm{cosec}x)}^2+576$$$$minf\left(x\right)is576$$$$$$$$or$$$$f\left(x\right)=256\sin {}^2\left(x\right)+324\mathrm{cosec}{}^2\left(x\right)$$$$⩾2\sqrt{256\sin {}^{2}x.324\mathrm{cosec}{}^{2}x}$$$$=2\left(16\right)\left(18\right)$$$$=576$$$$minf\left(x\right)is576$$

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