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Question Number 88040 by peter frank last updated on 08/Apr/20

$$Findthemaxandmin$$$$offunction$$$$\frac{a+b\sin x}{b+a\sin x}$$$$whereb>a>0inthe$$$$interval0⩽x⩽2\pi .sketch$$$$a=4andb=5$$

Commented by john santu last updated on 08/Apr/20

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{bsin}\mathrm{x}+\mathrm{a}}{\mathrm{asin}\mathrm{x}+\mathrm{b}}$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\frac{({\mathrm{b}}^2-{\mathrm{a}}^2)\cos \mathrm{x}}{{(\mathrm{asin}\mathrm{x}+\mathrm{b})}^2}=0$$$$({\mathrm{b}}^2-{\mathrm{a}}^2)\ne 0,\mathrm{then}\cos \mathrm{x}=0$$$$\cos \mathrm{x}=0\{\begin{array}{l}{\mathrm{x}}_1=\frac{\pi }{2}\\ \mathrm{x}{}_2=\frac{3\pi }{2}\end{array}$$$$\mathrm{f}\left({\mathrm{x}}_1\right)=\frac{\mathrm{b}+\mathrm{a}}{\mathrm{a}+\mathrm{b}}=1\leftarrow \max$$$$\mathrm{f}\left({\mathrm{x}}_2\right)=\frac{\mathrm{a}-\mathrm{b}}{\mathrm{b}-\mathrm{a}}=-1\leftarrow \min$$

Commented by peter frank last updated on 15/Apr/20

$$thankyou$$

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