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Question Number 144663 by mathdanisur last updated on 27/Jun/21

x∈(0;π) and (a;b) real numbers fixed. Find the range of function: g(x)= (((1+a^2 +cot^2 x)∙(1+b^2 +cot^2 x))/(1 + cot^2 x))

$${x}\in\left(\mathrm{0};\pi\right)\:{and}\:\left({a};{b}\right)\:{real}\:{numbers}\:{fixed}. \\ $$$${Find}\:{the}\:{range}\:{of}\:{function}: \\ $$$${g}\left({x}\right)=\:\frac{\left(\mathrm{1}+{a}^{\mathrm{2}} +{cot}^{\mathrm{2}} {x}\right)\centerdot\left(\mathrm{1}+{b}^{\mathrm{2}} +{cot}^{\mathrm{2}} {x}\right)}{\mathrm{1}\:+\:{cot}^{\mathrm{2}} {x}} \\ $$

Answered by mindispower last updated on 27/Jun/21

(a^2 +(1/(sin^2 (x))))((1/(sin^2 (x)))+b^2 ).sin^2 (x) =(a^2 +(1/(sin^2 (x))))(1+b^2 sin^2 (x)) cauchy shwartz≥(a+b)^2 min

$$\left({a}^{\mathrm{2}} +\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left({x}\right)}\right)\left(\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left({x}\right)}+{b}^{\mathrm{2}} \right).{sin}^{\mathrm{2}} \left({x}\right) \\ $$$$=\left({a}^{\mathrm{2}} +\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left({x}\right)}\right)\left(\mathrm{1}+{b}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({x}\right)\right) \\ $$$${cauchy}\:{shwartz}\geqslant\left({a}+{b}\right)^{\mathrm{2}} \:\:{min} \\ $$$$ \\ $$

Commented by mathdanisur last updated on 27/Jun/21