已知方程x²-2mx+4m²-6=0﹙m∈R﹚的两个实根为αβ求﹙α-1﹚²+﹙β-1﹚²的最大值和最小值
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![已知方程x²-2mx+4m²-6=0﹙m∈R﹚的两个实根为αβ求﹙α-1﹚²+﹙β-1﹚²的最大值和最小值](/uploads/image/z/2491987-67-7.jpg?t=%E5%B7%B2%E7%9F%A5%E6%96%B9%E7%A8%8Bx%26%23178%3B-2mx%2B4m%26%23178%3B-6%3D0%EF%B9%99m%E2%88%88R%EF%B9%9A%E7%9A%84%E4%B8%A4%E4%B8%AA%E5%AE%9E%E6%A0%B9%E4%B8%BA%CE%B1%CE%B2%E6%B1%82%EF%B9%99%CE%B1-1%EF%B9%9A%26%23178%3B%2B%EF%B9%99%CE%B2-1%EF%B9%9A%26%23178%3B%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC)
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已知方程x²-2mx+4m²-6=0﹙m∈R﹚的两个实根为αβ求﹙α-1﹚²+﹙β-1﹚²的最大值和最小值
已知方程x²-2mx+4m²-6=0﹙m∈R﹚的两个实根为αβ求﹙α-1﹚²+﹙β-1﹚²的最大值和最小值
已知方程x²-2mx+4m²-6=0﹙m∈R﹚的两个实根为αβ求﹙α-1﹚²+﹙β-1﹚²的最大值和最小值
判别式=4m²-16m²+24≥0
24≥12m²
m²≤2
-√2≤m≤√2
原式=α²-2α+1+β²-2β+1=(α+β)²-2αβ-2(α+β)+2=4m²-8m²+12-4m+2
=-(4m²+4m-14)=-4(m²+m-7/2)=-4[(m+1/2)²-1/4-14/4]=-4(m+1/2)²+15
当m=-1/2,原式最大值为15
当m=√2,原式最小值=-4(2+√2+1/4)+15=-9-4√2+15=6-4√2