已知函数f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4在0=

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已知函数f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4在0=
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已知函数f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4在0=
已知函数f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4在0=

已知函数f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4在0=
cosx=sinx是取得最大值,m=1

求二阶导数就行了.图我也画出来了.

f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4
设s=sinx,c=cosx
f(x)=2(s^4+c^4)+m(s+c)^4
=2[(s^2+c^2)^2-2s^2c^2]+m(1+4sc+4s^2c^2)
设t=sc=sin2x/2∈[0,1/2]
f(x)=2[1-2t^2]+m(1+4t+4t^2)
=2-4t^2+m+4mt+4mt^2
接下来就是求导,讨论最值了
m=1

f(x)=2(sin^4 x+cos^4 x)+m(sin^x+cosx)^4
f'(x)=8(sin^3 x cosx-cos^3 x sinx)+4m(sinx+cosx)^3(-sinx+cosx)
=8(sin^2x-cos^2x)sinx cosx-4m(sinx+cosx)^2(sin^2x-cos^2x)
=-4cos2x (sin2x-m(1-sin2x))
cos2x=0
x=pi/4 sinx=cosx
f(x)max=2(1/4+1/4)+4m=5
m=1