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sb688.cn:【金版优课】高中数学人教B版选修2-2课时作业:1.3.1 利用导数判断函数的单调性(1)(含答案解析)

第一章 §1.3 课时作业 6 一、选择题 1.下列函数中,在(0,+∞)内为增函数的是( A. y=sin x C. y=x -x 3 2 ) B. y=xe x D. y=-x+ln(1+x) 解析:y=xex,则 y′=ex+xex=ex(1+x)在(0,+∞)上恒大于 0. 答案:B 2.若函数 y=f(x)的导函数 在区间[a,b]上是增函数,则函数 y=f(x)在区间[a,b]上的图 ... 象可能是( ) 解析:∵y=f(x)的导函数在区间[a,b]上是增函数,则函数 f(x)图象上的点的切线斜率 是递增的. 答案:A 3.已知 f(x)=2cos2x+1,x∈(0,π),则 f(x)的单调递增区间是( A.(π,2π) π C.( ,π) 2 B.(0,π) π D.(0, ) 2 ) 解析:∵f(x)=2cos2x+1=2+cos2x,x∈(0,π), ∴f′(x)=-2sin2x. 令 f′(x)>0,则 sin2x<0. 又 x∈(0,π),∴0<2x<2π. π ∴π<2x<2π,即 <x<π. 2 答案:C 4.设函数 f(x)在定义域内可导,y=f(x)的图象如图所示,则导函数 y=f′(x)可能为( ) 解析:由函数的图象知:当 x<0 时,函数单调递增,导数始终为正;当 x>0 时,函数先 增后减再增,导数先正后负再正,对照选项,应选 D. 答案:D 二、填空题 5.函数 f(x)=x3+x2-5x-5 的单调递增区间是__________. 5 解析:令 y′=3x2+2x-5>0,得 x<- 或 x>1. 3 5 答案:(-∞,- )∪(1,+∞) 3 6.函数 y=ln(x2-x-2)的递减区间为__________. 解析: ∵f′(x)= 2x-1 2x-1 1 , 由 f′(x)= 2 <0, 得 x<-1 或 <x<2, 而函数的定义域为(- 2 x2-x-2 x -x-2 ∞,-1)∪(2,+∞),故递减区间为(-∞,-1). 答案:(-∞,-1) 1 7.设函数 f(x)=x(ex-1)- x2,则 f(x)的单调递增区间是 ________,单调递减区间是 2 ________. 解析:f′(x)=ex-1+xex-x=(ex-1)(x+1). 当 x∈(-∞,-1)时,f′(x)>0; 当 x∈(-1,0)时,f′(x)<0; 当 x∈(0,+∞)时,f′(x)>0. 故 f(x)在(-∞,-1),(0,+∞)上单调递增, 在(-1,0)上单调递减. 答案:(-∞,-1)和(0,+∞) (-1,0) 三、解答题 8.证明:函数 f(x)=lnx+x 在其定义域内为单调递增函数. 证明:函数的定义域为{x|x>0}, 1 又 f′(x)=(lnx+x)′= +1, x 当 x>0 时,f′(x)>1>0, 故 y=lnx+x 在其定义域内为单调递增函数. 9.已知函数 f(x)=x2· ex 1+ax3+bx2,且 x=-2 和 x=1 是 f′(x)=0 的两根. - (1)求 a,b 的值: (2)求 f(x)的单调区间. 解:(1)因为 f′(x)=ex 1(2x+x2)+3ax2+2bx - =xex 1(x+2)+x(3ax+2b), - 又 x=-2 和 x=1 为 f′(x)=0 的两根, 所以 f′(-2)=f′(1)=0. ?-6a+2b=0 ? 故有? , ?3+3a+2b=0 ? 1 解方程组得 a=- ,b=-1. 3 1 (2)因为 a=- ,b=-1, 3 ∴f′(x)=x(x+2)(ex 1-1). - 令 f′(x)=0 得 x1=-2,x2=0,x3=1. 当 x∈(-2,0)∪(1,+∞)时,f′(x)>0; 当 x∈(-∞,-2)∪(0,1)时,f′(x)<0, ∴f(x)的单调递增区间为(-2,0)和(1,+∞),单调递减区间为(-∞,-2)和(0,1).

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