Prove that
a. -4 ≤(cos2x+3 sin x) ≤ (17/8)

b. abcd>81 (s-a)(s-b)(s-c)(s-d)

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Please find below the solution to the asked query:

$$\begin{gathered} \left(\mathrm{a}\right) \\ \mathrm{Let}, \\ \mathrm{f}\left(\mathrm{x}\right)=\cos 2\mathrm{x}+3\mathrm{sinx} \\ \mathrm{Using} \mathrm{identity} \cos 2\mathrm{\alpha }=1-2{\sin }^2\mathrm{\alpha } \\ \Rightarrow \mathrm{f}\left(\mathrm{x}\right)=1-2{\sin }^2\mathrm{x}+3\mathrm{sinx} ; \mathrm{equation}\left(\mathrm{i}\right) \\ \mathrm{Differentiating} \mathrm{with} \mathrm{respect} \mathrm{to} \mathrm{x} \mathrm{we} \mathrm{get}, \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=-2.2.\mathrm{sinx}.\mathrm{cosx}+3\mathrm{cosx} \\ \Rightarrow \mathrm{f}\text{'}\left(\mathrm{x}\right)=-4\mathrm{sinx}.\mathrm{cosx}+3\mathrm{cosx} \\ \Rightarrow \mathrm{f}\text{'}\left(\mathrm{x}\right)=\mathrm{cosx}\left(3-4\mathrm{sinx}\right) \\ \mathrm{For} \mathrm{maxima} \mathrm{or} \mathrm{minima} \mathrm{f}\text{'}\left(\mathrm{x}\right)=0 \\ \Rightarrow \mathrm{cosx}\left(3-4\mathrm{sinx}\right)=0 \\ \mathrm{Either} \mathrm{cosx}=0 \mathrm{or} 3-4\mathrm{sinx}=0 \\ \mathrm{when} \cos \mathrm{x}=0 \\ \sin \mathrm{x}=\pm 1 \\ \mathrm{Put} \sin \mathrm{x}=1 \mathrm{in} \mathrm{equation}\left(\mathrm{i}\right), \mathrm{we} \mathrm{get}, \\ \mathrm{f}\left(\mathrm{x}\right)=1-2+3 \\ \mathbf{\Rightarrow }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}\mathbf{2}\mathbf{ }\mathbf{;}\mathbf{ }\mathbf{case}\left(\mathbf{i}\right) \\ \mathrm{Put} \sin \mathrm{x}=-1 \mathrm{in} \mathrm{equation}\left(\mathrm{i}\right), \mathrm{we} \mathrm{get}, \\ \mathrm{f}\left(\mathrm{x}\right)=1-2-3 \\ \mathbf{\Rightarrow }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{ }\mathbf{;}\mathbf{ }\mathbf{case}\left(\mathbf{ii}\right) \\ \mathrm{When} 3-4\mathrm{sinx}=0 \\ \Rightarrow \sin \mathrm{x}=\frac{3}{4} \\ \mathrm{Put} \sin \mathrm{x}=\frac{3}{4} \mathrm{in} \mathrm{equation}\left(\mathrm{i}\right), \mathrm{we} \mathrm{get}, \\ \Rightarrow \mathrm{f}\left(\mathrm{x}\right)=1-2{\left(\frac{3}{4}\right)}^2+3\left(\frac{3}{4}\right) \\ =1-\frac{9}{8}+\frac{9}{4} \\ =\frac{8-9+18}{8} \\ \mathbf{\Rightarrow }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}\frac{\mathbf{17}}{\mathbf{8}}\mathbf{ }\mathbf{;}\mathbf{case}\left(\mathbf{iii}\right) \\ \mathrm{From} \mathrm{case}\left(\mathrm{i}\right), \mathrm{case}\left(\mathrm{ii}\right), \mathrm{case}\left(\mathrm{iii}\right), \mathrm{we} \mathrm{get}, \\ -4\le \mathrm{f}\left(\mathrm{x}\right)\le \frac{17}{8} \\ \mathbf{\Rightarrow }\mathbf{-}\mathbf{4}\mathbf{\le }\mathbf{cos}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{sinx}\mathbf{\le }\frac{\mathbf{17}}{\mathbf{8}}\mathbf{ }\left(\mathbf{Hence}\mathbf{ }\mathbf{proved}\right) \\ \mathrm{As} \mathrm{for} \mathrm{second} \mathrm{part} \mathrm{of} \mathrm{your} \mathrm{question}, \mathrm{please} \mathrm{provide} \mathrm{the} \mathrm{relation} \mathrm{between} \mathrm{s} \mathrm{and} \left(\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\right) \mathrm{so} \mathrm{that} \mathrm{we} \mathrm{can} \mathrm{hep} \mathrm{you} \mathrm{further}. \end{gathered}$$

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