Toán Lớp 11: y=cox ²x +2cos2x. Tìm max- min

Question

Toán Lớp 11:  y=cox ²x +2cos2x. Tìm max- min

khanhlanchau · Answer

~rai~        (y= cos^2x+2 cos2x   quad= dfrac{1+ cos2x}{2}+2 cos2x   quad= dfrac{1}{2}(1+ cos2x+4 cos2x)   quad= dfrac{1}{2}(1+5 cos2x)   text{Ta c&oacute;:}-1 le cos2x le 1   Leftrightarrow -5 le5 cos2x le 5   Leftrightarrow -4 le1+5 cos2x le 6   Leftrightarrow -2 le dfrac{1}{2}(1+5 cos2x) le 3   Leftrightarrow -2 le y le3.  +)Min_y=-2 Leftrightarrow  cos2x=-1   Leftrightarrow 2x= pi+k2 pi   Leftrightarrow x= dfrac{ pi}{2}+k pi.(k in mathbb{Z})  +)Max_y=1 Leftrightarrow  cos2x=1   Leftrightarrow 2x=k2 pi   Leftrightarrow x=k pi.(k in mathbb{Z})   text{Vậy Min}_y=-2 text{ khi x=} dfrac{ pi}{2}+k pi;   text{Max}_y=3 text{ khi x=}k pi.(k in mathbb{Z}) )

hienthucthu97 · Answer

Giải đáp:       Lời giải và giải thích chi tiết:   $ y = cos&sup2;x + 2cos2x $   $ = cos&sup2;x + 2(2cos&sup2;x - 1) = 5cos&sup2;x - 2 $   Do $ 0 &le; cos&sup2;x &le; 1 &hArr; 0 &le; 5cos&sup2;x &le; 5 $   $ &hArr; - 2 &le; 5cos&sup2;x - 2 &le; 3 &hArr; - 2 &le; y &le; 3$   $ &rArr; Miny = - 2 &hArr; cosx = 0 &hArr; x = (2k + 1) dfrac{&pi;}{2}$   $ &rArr; Maxy = 3 &hArr; cosx = &plusmn;1 &hArr; x = k&pi; $