设集合A={(x,y)丨y=x^2+ax+2},B={(x,y)丨y=x+1,0小于等于x小于等于2},A∩B不等于空集,求实数a的取值范围RT
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![设集合A={(x,y)丨y=x^2+ax+2},B={(x,y)丨y=x+1,0小于等于x小于等于2},A∩B不等于空集,求实数a的取值范围RT](/uploads/image/z/10200626-26-6.jpg?t=%E8%AE%BE%E9%9B%86%E5%90%88A%3D%7B%28x%2Cy%29%E4%B8%A8y%3Dx%5E2%2Bax%2B2%7D%2CB%3D%7B%28x%2Cy%29%E4%B8%A8y%3Dx%2B1%2C0%E5%B0%8F%E4%BA%8E%E7%AD%89%E4%BA%8Ex%E5%B0%8F%E4%BA%8E%E7%AD%89%E4%BA%8E2%7D%2CA%E2%88%A9B%E4%B8%8D%E7%AD%89%E4%BA%8E%E7%A9%BA%E9%9B%86%2C%E6%B1%82%E5%AE%9E%E6%95%B0a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4RT)
设集合A={(x,y)丨y=x^2+ax+2},B={(x,y)丨y=x+1,0小于等于x小于等于2},A∩B不等于空集,求实数a的取值范围RT
设集合A={(x,y)丨y=x^2+ax+2},B={(x,y)丨y=x+1,0小于等于x小于等于2},A∩B不等于空集,求实数a的取值范围
RT
设集合A={(x,y)丨y=x^2+ax+2},B={(x,y)丨y=x+1,0小于等于x小于等于2},A∩B不等于空集,求实数a的取值范围RT
两集合有交集,则数对满足:x+1=x^2+ax+2
整理,得
x^2+(a-1)x+1=0
交集非空就是这个方程有实根,且根在[0,2]上.
判别式:
(a-1)^2-4≥0
a≥3或a≤-1
a≥3时,函数y=x^2+(a-1)x+1对称轴横坐标≤-1,函数在[0,2]上单调增,又x=0时,y=1>0,不满足题意.
a≤-1时,函数y=x^2+(a-1)x+1对称轴横坐标≥1,函数在[0,2]上单调减,又x=0时,y=1>0,因此,只要x=2时,x^2+(a-1)x+1≤0
4+2(a-1)+1≤0
解得a≤-3/2
a的取值范围为(-∞.-3/2]