Question Number 136437 by liberty last updated on 22/Mar/21
$${Find}\:{max}\:{and}\:{min}\:{value}\:{of}\: \\ $$$${y}\:=\:\mathrm{sin}\:^{\mathrm{2}} {x}+\:\mathrm{cos}\:^{\mathrm{4}} {x}\: \\ $$
Answered by rs4089 last updated on 22/Mar/21
$${y}=\mathrm{1}−{cos}^{\mathrm{2}} {x}+{cos}^{\mathrm{4}} {x} \\ $$$${y}=\frac{\mathrm{3}}{\mathrm{4}}+\left({cos}^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${y}_{{min}.} =\frac{\mathrm{3}}{\mathrm{4}}+\mathrm{0}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$${y}_{{max}.} =\frac{\mathrm{3}}{\mathrm{4}}+\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} =\mathrm{1} \\ $$