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Question Number 213643 by Abdullahrussell last updated on 12/Nov/24
 Find the maximum value of   3sin^2 x−8cosx+5=?
$$\:{Find}\:{the}\:{maximum}\:{value}\:{of} \\ $$$$\:\mathrm{3}{sin}^{\mathrm{2}} {x}−\mathrm{8}{cosx}+\mathrm{5}=? \\ $$
Answered by Berbere last updated on 12/Nov/24
sin^2 (x)=1−cos^2 (x)  t=cos(x);t∈[−1,1]  study t→f(t)=3(1−t^2 )−8t+5 t
$${sin}^{\mathrm{2}} \left({x}\right)=\mathrm{1}−{cos}^{\mathrm{2}} \left({x}\right) \\ $$$${t}={cos}\left({x}\right);{t}\in\left[−\mathrm{1},\mathrm{1}\right] \\ $$$${study}\:{t}\rightarrow{f}\left({t}\right)=\mathrm{3}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)−\mathrm{8}{t}+\mathrm{5}\:{t} \\ $$
Answered by golsendro last updated on 12/Nov/24
  F(x)= 3(1−cos^2 x)−8cos x+5    F(x)= −3cos^2 x−8cos x+ 8    cos x=1⇒F_1 =−3−8+8 =−3    cos x=−1⇒F_2 =−3+8+8= 13    cos x=−(((−8))/(2.(−3)))= −(4/3) (rejected)    ∴ max value = 13
$$\:\:\mathrm{F}\left(\mathrm{x}\right)=\:\mathrm{3}\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\right)−\mathrm{8cos}\:\mathrm{x}+\mathrm{5} \\ $$$$\:\:\mathrm{F}\left(\mathrm{x}\right)=\:−\mathrm{3cos}\:^{\mathrm{2}} \mathrm{x}−\mathrm{8cos}\:\mathrm{x}+\:\mathrm{8} \\ $$$$\:\:\mathrm{cos}\:\mathrm{x}=\mathrm{1}\Rightarrow\mathrm{F}_{\mathrm{1}} =−\mathrm{3}−\mathrm{8}+\mathrm{8}\:=−\mathrm{3} \\ $$$$\:\:\mathrm{cos}\:\mathrm{x}=−\mathrm{1}\Rightarrow\mathrm{F}_{\mathrm{2}} =−\mathrm{3}+\mathrm{8}+\mathrm{8}=\:\mathrm{13} \\ $$$$\:\:\mathrm{cos}\:\mathrm{x}=−\frac{\left(−\mathrm{8}\right)}{\mathrm{2}.\left(−\mathrm{3}\right)}=\:−\frac{\mathrm{4}}{\mathrm{3}}\:\left(\mathrm{rejected}\right) \\ $$$$\:\:\therefore\:\mathrm{max}\:\mathrm{value}\:=\:\mathrm{13} \\ $$
Answered by a.lgnaoui last updated on 12/Nov/24
 let  f(x)=3sin^2 x−8cos x+5    { ((Max(3sin^2  x,−8cos x)=+8  [x=(2k+1)π])),(( ⇒  f(x)≤13         )) :}     So     Max(f(x))=13          ^�
$$\:\mathrm{let}\:\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{8cos}\:\mathrm{x}+\mathrm{5} \\ $$$$\:\begin{cases}{\mathrm{Max}\left(\mathrm{3sin}^{\mathrm{2}} \:\mathrm{x},−\mathrm{8cos}\:\mathrm{x}\right)=+\mathrm{8}\:\:\left[\mathrm{x}=\left(\mathrm{2k}+\mathrm{1}\right)\pi\right]}\\{\:\Rightarrow\:\:\mathrm{f}\left(\mathrm{x}\right)\leqslant\mathrm{13}\:\:\:\:\:\:\:\:\:}\end{cases} \\ $$$$\:\:\:\mathrm{So}\:\:\:\:\:\boldsymbol{\mathrm{Max}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\right)=\mathrm{13} \\ $$$$\:\:\:\:\:\:\:\hat {\:}\: \\ $$

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