问题 解答题

已知函数f(x)对任意实数x均有f(x)=kf(x+2),其中常数k为负数,且f(x)在区间[0,2]上有表达式f(x)=x(x-2)。

(I)求f(-1),f(2.5)的值;

(Ⅱ)写出f(x)在[-3,3]上的表达式,并讨论函数f(x)在[ -3,3]上的单调性;

(Ⅲ)求出f(x)在[-3,3]上的最小值与最大值,并求出相应的自变量的取值。

答案

解:(Ⅰ)f(-1)=kf(1)=-k

∵f(0.5)=kf(2.5)

(Ⅱ)∵对任意实数x,f(x)=kf(x+2)

∴f(x-2)=kf(x)

∴f(x)=

当-2≤x<0,0≤x+2<2,f(x)=kf(x+2)=kx(x+2);

当-3≤x<-2,-1≤x+2<0,f(x)=kf(x+2)=k2(x+2)(x+4);

当2≤x≤3时,0≤x-2≤1,

∴k<0

∴f(x)在[-3,-1]与[1,3]上为增函数,在[-1,1] 上为减函数;

(Ⅲ)由函数f(x)在[-3,3]上的单调性可知f(x)在x=-3或x=1处取得最小值f(-3)=-k2或f(1)=-1,而在x=-1或x=3处取得最大值,f(-1)=-k或f(3)=-

故有①k<-1时,f(x)在x=-3处取得最小值f(-3)=-k2,在x=-1处取得最大值,f(-1)=-k;

②k=-1时,f(x)在x=-3与x=l处取得最小值f(-3)= f(1)=-1,在x=-1与x=3处取得最大值f(-1)=f(3)=1

③-1<k<0时,f(x)在x=1处取得最小值f(1)=-1,在x=3处取得最大值

句型转换

句型转换。

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2. She put out the fire with a blanket. (对划线部分提问)

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3. She was in hospital for two months.(对划线部分提问) 

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4. Our teacher is recommending Amy for the Best Sports Player Award. (对划线部分提问) 

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5. His friend will be away from here for two months. (对划线部分提问) 

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6. He could not speak Chinese before. (用now代替before改写句子成肯定句)

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7. She found a purse on her way home. (用Tom代替she改写句子)

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8. Take a seat, please. (同义句)

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9. They are not happy. (否定句)

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10. Wang Fang stopped a fire from burning and she saved her neighbor. (同义句)

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11. She is good at playing tennis. (同义句)

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12. The teacher asked us, "How do you solve this problem?" (改为间接引语)

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13. We should try to study best. (同义句)

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