Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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

Find the max and min  of function  ((a+bsin x)/(b+asin x))  where b>a>0 in the   interval 0≤x≤2π.sketch  a=4 and b=5

$${Find}\:{the}\:{max}\:{and}\:{min} \\ $$ $${of}\:{function} \\ $$ $$\frac{{a}+{b}\mathrm{sin}\:{x}}{{b}+{a}\mathrm{sin}\:{x}} \\ $$ $${where}\:{b}>{a}>\mathrm{0}\:{in}\:{the}\: \\ $$ $${interval}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\pi.{sketch} \\ $$ $${a}=\mathrm{4}\:{and}\:{b}=\mathrm{5} \\ $$

Commented byjohn santu last updated on 08/Apr/20

f(x) = ((bsin x+a)/(asin x+b))  f′(x) = (((b^2 −a^2 )cos x)/((asin x+b)^2 )) = 0  (b^2 −a^2 ) ≠0 , then cos x = 0  cos x = 0  { ((x_1 =(π/2))),((x _2 = ((3π)/2))) :}  f(x_1 ) = ((b+a)/(a+b)) = 1 ←max  f(x_2 ) = ((a−b)/(b−a)) = −1 ←min

$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{bsin}\:\mathrm{x}+\mathrm{a}}{\mathrm{asin}\:\mathrm{x}+\mathrm{b}} \\ $$ $$\mathrm{f}'\left(\mathrm{x}\right)\:=\:\frac{\left(\mathrm{b}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{asin}\:\mathrm{x}+\mathrm{b}\right)^{\mathrm{2}} }\:=\:\mathrm{0} \\ $$ $$\left(\mathrm{b}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\:\neq\mathrm{0}\:,\:\mathrm{then}\:\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{0} \\ $$ $$\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{0}\:\begin{cases}{\mathrm{x}_{\mathrm{1}} =\frac{\pi}{\mathrm{2}}}\\{\mathrm{x}\:_{\mathrm{2}} =\:\frac{\mathrm{3}\pi}{\mathrm{2}}}\end{cases} \\ $$ $$\mathrm{f}\left(\mathrm{x}_{\mathrm{1}} \right)\:=\:\frac{\mathrm{b}+\mathrm{a}}{\mathrm{a}+\mathrm{b}}\:=\:\mathrm{1}\:\leftarrow\mathrm{max} \\ $$ $$\mathrm{f}\left(\mathrm{x}_{\mathrm{2}} \right)\:=\:\frac{\mathrm{a}−\mathrm{b}}{\mathrm{b}−\mathrm{a}}\:=\:−\mathrm{1}\:\leftarrow\mathrm{min} \\ $$

Commented bypeter frank last updated on 15/Apr/20

thank you

$${thank}\:{you} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com