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Question Number 153870 by physicstutes last updated on 11/Sep/21

$$\mathrm{find}\mathrm{the}\mathrm{minimum}\mathrm{and}\mathrm{maximum}\mathrm{value}$$$$\mathrm{of}\frac{5}{f\left(\theta \right)+3}\mathrm{where}f\left(\theta \right)=8\cos \theta -15\sin \theta$$

Commented by mr W last updated on 12/Sep/21

$$therearenominimumandno$$$$maximum.thereareonlylocal$$$$minimumandlocalmaximum.$$

Commented by physicstutes last updated on 12/Sep/21

why sir?

Commented by mr W last updated on 12/Sep/21

$$f\left(\theta \right)=8\cos \theta -15\sin \theta =17\cos (\theta +\alpha )$$$$-17⩽f\left(\theta \right)⩽17$$$$thatmeansf\left(\theta \right)canbe-3,$$$$whenf\left(\theta \right)\to -3^{-},\frac{5}{f\left(\theta \right)+3}\to -\mathrm{\infty }\Rightarrow nominimum$$$$whenf\left(\theta \right)\to -3^{+},\frac{5}{f\left(\theta \right)+3}\to +\mathrm{\infty }\Rightarrow nomaximum$$

Answered by liberty last updated on 11/Sep/21

$$g\left(\theta \right)=\frac{5}{f\left(\theta \right)+3}=\frac{5}{17\cos (\theta -\vartheta )+3}$$$$\Rightarrow 17\cos (\theta -\vartheta )+3=\frac{5}{g\left(\theta \right)}$$$$\Rightarrow 17\cos (\theta -\vartheta )=\frac{5}{g\left(\theta \right)}-3$$$$weknowthat-17⩽17\cos (\theta -\vartheta )⩽17$$$$-17⩽\frac{5}{g\left(\theta \right)}-3⩽17$$$$-14⩽\frac{5}{g\left(\theta \right)}⩽20$$$$-\frac{14}{5}⩽\frac{1}{g\left(\theta \right)}⩽4$$$$\frac{1}{4}⩽g\left(\theta \right)<\mathrm{\infty }$$$$minimum=\frac{1}{4}$$$$hasnomaximum$$

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