函數(shù)f(x)=(a-1)x2+2ax+1在區(qū)間(1,2)上是增函數(shù),則實數(shù)a的取值范圍是 .
【答案】
分析:當(dāng)a=1時,f(x)=2x+1在區(qū)間(1,2)上是增函數(shù).當(dāng)a>1時,對稱軸x=
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≤1,解得a>1.當(dāng)a<1時,對稱軸x=
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≥2,解得
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.由此能求出實數(shù)a的取值范圍.
解答:解:當(dāng)a=1時,f(x)=2x+1在區(qū)間(1,2)上是增函數(shù).
當(dāng)a>1時,由題意知,對稱軸x=
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≤1,
解得a>1.
當(dāng)a<1時,由題意知,對稱軸x=
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≥2,
解得
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.
綜上所述,實數(shù)a的取值范圍是[
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.
故答案為:[
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.
點評:本題是一類考查對二次函數(shù)系數(shù)討論的非常典型的試題,一定要熟悉其方法:1.當(dāng)a=0時,函數(shù)是一次函數(shù),明顯在規(guī)定區(qū)間是增函數(shù),符合題意.2.當(dāng)a不等于0時,函數(shù)是二次函數(shù),這時候一定要注意數(shù)形結(jié)合分析題目(對于函數(shù)、立體幾何和解析幾何數(shù)形結(jié)合是非常必要的,切記)當(dāng)a>0時,函數(shù)開口向上,通過畫圖可以發(fā)現(xiàn)只有當(dāng)對稱軸在1/2左側(cè)的時候,才滿足題意,故可求得a的取值范圍.同理可得,當(dāng)a<0時,只有當(dāng)對稱軸在1右側(cè)的時候,才滿足題意,故可求得a的取值范圍.綜合以上2種情況可得a的取值范圍.