[-1,
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]
分析:本題屬于線性規(guī)劃中的延伸題,對于可行域不要求線性目標函數(shù)的最值,而是求可行域內的點與(4,1)構成的直線的斜率問題,求出斜率的取值范圍,從而求出目標函數(shù)的取值范圍.
解答:
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解:由于z=
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=
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,
由x,y滿足約束條件
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所確定的可行域如圖所示,
考慮到
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可看成是可行域內的點與(4,1)構成的直線的斜率,
結合圖形可得,
當Q(x,y)=A(3,2)時,z有最小值1+2×
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=-1,
當Q(x,y)=B(-3,-4)時,z有最大值 1+2×
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=
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,
所以-1≤z≤
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.
故答案為:[-1,
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]
點評:本題考查線性規(guī)劃問題,難點在于目標函數(shù)幾何意義,近年來高考線性規(guī)劃問題高考數(shù)學考試的熱點,數(shù)形結合是數(shù)學思想的重要手段之一,是連接代數(shù)和幾何的重要方法.