2022~23年度考前模拟演练卷(四)文科综合试题 答案

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22.(满分12分)已知函数f(x)=ar+l)2-e*(aeR):()当a=时，判断函数fw的单调性：2若有两个极值点6G<.证明：<)<-日第7真共8真【解折】山当a=时，fx)=x+-c,f()=x+1-e,…1分记g(x)=x+1-e,则g(x）=1-e,由g'(x)=1-e>0得x<0,由g'(x)=1-e<0得x>0,·g(x)即f'(x)在区间(-o,0)上单调递增，在区间(0，+）上单调递减，1■■■■e后e有有e44444年4n8■■■ER81(x)=f'(0)=0,对xsR,f(x)s0,3分∴f(x)在R上单调递减：4分(2:f(x)有两个极值点，关于x的方程'(x)=2x+)-心=0有两个根x,设p(x)=2a(x+l)-e,则p'(x)=2a-e,5分当a≤0时，9(x)=20-c<0,即(x)即f(x)在k上单调递减，二"(x)=0最多有一个根，不符题意：…6分当a>0时，由(x)>0,得x 1n2a,p(x)即f'(x)在区间(-o,ln2a)上单调递增，在区间(ln2a,+o)上单调递减.且当x→-o时，'(x)→-0，当x→+时，f"(x)→-0，7分要使f"(x)=0有两个不同的根，必有f(x)m=f"(n2a)=2an2a+12a=2alh2a>0,解得a>。8分r(-)=-1<0,f(0)=2a-1>0,-1

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