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Question Number 126098 by ZiYangLee last updated on 17/Dec/20

$$\mathrm{Prove}\mathrm{that}$$$$27({\sin }^4\alpha +{2\cos }^4\alpha )⩾16{\left(\sin 2\alpha \right)}^2$$

Answered by MJS_new last updated on 17/Dec/20

$$\sin 2\alpha =2\sin \alpha \cos \alpha$$$$27(s^4+2c^4)⩾{64\mathrm{s}}^2{\mathrm{c}}^2$$$$c=\sqrt{1-s^2}$$$$27(3s^4-4s^2+2)⩾64s^2(1-s^2)$$$$s^4-\frac{172}{145}{s}^2+\frac{54}{145}⩾0$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{easy}\mathrm{to}\mathrm{show}\mathrm{this}\mathrm{has}\mathrm{no}\mathrm{real}\mathrm{zeros}$$$$\mathrm{and}\mathrm{for}s=0{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{true}\Rightarrow \mathrm{always}\mathrm{true}$$

Commented by talminator2856791 last updated on 17/Dec/20

$$\mathrm{very}\mathrm{nice}.\mathrm{quadratic}\mathrm{is}\mathrm{perfect}.$$

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