已知x-y=a,z-y=10,求x^2+y^2+z^2-xy-yx-zx的最小值
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已知x-y=a,z-y=10,求x^2+y^2+z^2-xy-yx-zx的最小值已知x-y=a,z-y=10,求x^2+y^2+z^2-xy-yx-zx的最小值已知x-y=a,z-y=10,求x^2+
已知x-y=a,z-y=10,求x^2+y^2+z^2-xy-yx-zx的最小值
已知x-y=a,z-y=10,求x^2+y^2+z^2-xy-yx-zx的最小值
已知x-y=a,z-y=10,求x^2+y^2+z^2-xy-yx-zx的最小值
由于x-y=a,z-y=10
得
x-z=a-10
并且由
x²+y²+z²-xy-yx-zx=1/2[(x-y)²+(y-z)²+(z-x)²]=[a²+10²+(10-a)²]/2
=(2a²-20a+200)/2=a²-10a+100=(a-5)²+75≥75.
即最小值为75.
=1/2(x-y)^2+(x-z)^2+(y-z)^2=1/2(a^2+(10-a)^2+10^2)
=a^2+100-10a=(a-5)^2+75>=75
答案为75
x-y=a
z-y=10
两个式子相减:x-z=a-10
x^2+y^2+z^2-xy-yz-zx
=0.5(x^2+y^2-2xy)+0.5(x^2+z^2-2zx)+0.5(y^2+z^2-2yz)
=0.5[(x-y)^2+(x-z)^2+(y-z)^2]
=0.5[a^2+(a-10)^2+100]
=a^2-10a+100
=(a-5)^2+75
所以最小值是75
这个老师教过。
2(x^2+y^2+z^2-xy-xz-zy)=(x-y)^2+(x-z)^2+(z-y)^2=a^2+(a-10)^2+10^2=最后结果你自己算去吧,那不不懂可以问我
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