若x-1=(y+1)/2=(z-2)/3,求x^2+y^2+z^2的最小值.
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若x-1=(y+1)/2=(z-2)/3,求x^2+y^2+z^2的最小值.
x^2+y^2+z^2=(x-1)^2+(y+1)^2+(z-2)^2+2x-1-2y-1+4z-4=14(x-1)^2+10(x-1)+6=14[(x-1)+5/14]^2-25/14+6,可知当x-1=-5/14时最小,即x=9/14,最小值为6-25/14=59/14
设(x-1)=(y+1)/2=(z-2)/3=k,则x=k+1,y=2k-1,z=3k+2,代入,得到一个二次函数,求其最小值。
当x=9/14时最小,=10-9/14=131/14,yz用x表示,化简为a(x+m)^2+n,最小或最大值就是n
由x-1=(y+1)/2=(z-2)/3
得到y=2x-3
z=3x-1
得到x^2+y^2+z^2 =x^2+(2x-3)^2+(3x-1)^2 =14*x^2-18x+13
当x= -(b/2a)= -(18/2*14)= -9/14时
x^2+y^2+z^2的值最小,是多少自己计算一下
令x-1=(y+1)/2=(z-2)/3=k,则x=k+1,y=2k-1,z=3k+2,
x^2+y^2+z^2=(k+1)²+(2k-1)²+(3k+2)²=14k²+10k+6
=14(k²+10k/14+25/196)+6-25/14=14(k+5/14)²+59/14,当k=-5/14时,取最小值59/14
令m=x^2+y^2+z^2
根据前面的式子,用x代替y,z.带入到m 中。
算出m=14x^2+18x+10
开口向上,算出的顶点就是所求的最小值
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