已知cosx+siny=1/2,求siny-cosx平方的最值,要具体过程!在线等!

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/21 02:57:24

cosx + siny = 1/2 (1)
Taking derivative to the both side w.r.t x
-sinx + cosy dy/dx =0
dy/dx = sinx/cosy
S = (siny - cosx)^2
= (cosx + siny)^2 - 4sinycosx
= 1/4 - 4sinycosx
dS/dx = -4[ -siny sinx + cosx cos y dy/dx ]
= -4[ - sinxsiny + sinxcosx ]
dS/dx = 0
sinx[ -siny + cosx ] =0
cosx = siny
or sinx =0
siny = cosx
from (1)
2siny = 1/2
siny = 1/4
or sinx=0 => x =0
from (1)
1+siny = 1/2
siny = -1/2
max S at siny = -1/2 ,cosx = 1
max S = 1/4 - 4(-1/2)(1)
= 3/8
minS = 1/4 - 4(1/4)(1/4)
= 3/16

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