Question Number 213643 by Abdullahrussell last updated on 12/Nov/24
$$\:{Find}\:{the}\:{maximum}\:{value}\:{of} \\ $$$$\:\mathrm{3}{sin}^{\mathrm{2}} {x}−\mathrm{8}{cosx}+\mathrm{5}=? \\ $$
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](https://www.tinkutara.com/question/Q213644.png)
$${sin}^{\mathrm{2}} \left({x}\right)=\mathrm{1}−{cos}^{\mathrm{2}} \left({x}\right) \\ $$$${t}={cos}\left({x}\right);{t}\in\left[−\mathrm{1},\mathrm{1}\right] \\ $$$${study}\:{t}\rightarrow{f}\left({t}\right)=\mathrm{3}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)−\mathrm{8}{t}+\mathrm{5}\:{t} \\ $$
Answered by golsendro last updated on 12/Nov/24
$$\:\:\mathrm{F}\left(\mathrm{x}\right)=\:\mathrm{3}\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\right)−\mathrm{8cos}\:\mathrm{x}+\mathrm{5} \\ $$$$\:\:\mathrm{F}\left(\mathrm{x}\right)=\:−\mathrm{3cos}\:^{\mathrm{2}} \mathrm{x}−\mathrm{8cos}\:\mathrm{x}+\:\mathrm{8} \\ $$$$\:\:\mathrm{cos}\:\mathrm{x}=\mathrm{1}\Rightarrow\mathrm{F}_{\mathrm{1}} =−\mathrm{3}−\mathrm{8}+\mathrm{8}\:=−\mathrm{3} \\ $$$$\:\:\mathrm{cos}\:\mathrm{x}=−\mathrm{1}\Rightarrow\mathrm{F}_{\mathrm{2}} =−\mathrm{3}+\mathrm{8}+\mathrm{8}=\:\mathrm{13} \\ $$$$\:\:\mathrm{cos}\:\mathrm{x}=−\frac{\left(−\mathrm{8}\right)}{\mathrm{2}.\left(−\mathrm{3}\right)}=\:−\frac{\mathrm{4}}{\mathrm{3}}\:\left(\mathrm{rejected}\right) \\ $$$$\:\:\therefore\:\mathrm{max}\:\mathrm{value}\:=\:\mathrm{13} \\ $$
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 ^](https://www.tinkutara.com/question/Q213649.png)
$$\:\mathrm{let}\:\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{8cos}\:\mathrm{x}+\mathrm{5} \\ $$$$\:\begin{cases}{\mathrm{Max}\left(\mathrm{3sin}^{\mathrm{2}} \:\mathrm{x},−\mathrm{8cos}\:\mathrm{x}\right)=+\mathrm{8}\:\:\left[\mathrm{x}=\left(\mathrm{2k}+\mathrm{1}\right)\pi\right]}\\{\:\Rightarrow\:\:\mathrm{f}\left(\mathrm{x}\right)\leqslant\mathrm{13}\:\:\:\:\:\:\:\:\:}\end{cases} \\ $$$$\:\:\:\mathrm{So}\:\:\:\:\:\boldsymbol{\mathrm{Max}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\right)=\mathrm{13} \\ $$$$\:\:\:\:\:\:\:\hat {\:}\: \\ $$