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高中数学(人教A版)选修2-2同步课后巩固:1-4 生活中的优化问题举例

1-4 生活中的优化问题举例
1.设底为等边三角形的直棱柱的体积为 V,那么其表面积最小时,底面边 长为( ) 3 B. 2V 3 D.2 V

3 A. V 3 C. 4V 答案 C

解析 设底面边长为 x,则表面积 3 4 3V 3 S= 2 x2+ x (x>0),S′= x2 (x3-4V). 3 令 S′=0,得唯一极值点 x= 4V. 2.用边长为 48 cm 的正方形铁皮做一个无盖的铁盒时,在铁皮的四角各截 去一个面积相等的小方形,然后把四边折起,就能焊成铁盒,所做的铁盒容积最 大时,在四角截去的正方形的边长为( A.6 C.10 答案 B 3.用长度为 l 的铁丝围成长方形,则围成的长方形的最大面积为( l2 A. 2 l2 C. 8 答案 D l-2x 解析 设长方形一边为 x,则另一边为 2 . l-2x l S=x· 2 =-x2+2x. l2 B. 4 l2 D.16 ) ) B.8 D.12

l l S′=-2x+2. 令 S′=0,得 x=4. l 2 l l l2 ∴S 最大=-(4) +2· 4=16. 4.设函数 f(x)=-x(x-a)2(x∈R,其中 a∈R). (1)当 a=1 时,求曲线 y=f(x)在点(2,f(2))处的切线方程; (2)当 a≠0 求函数 f(x)的极大值和极小值; (3)当 a>3 时,证明存在 k∈ [-1,0],使得不等式 f(k-cosx)≥f(k2-cos2x)对任意的 x∈R 恒成立. 解析 (1)当 a=1 时, f(x)=-x(x-1)2=-x3+2x2-x,得 f(2)=-2, f′(x)=-3x2+4x-1,f′(2)=-12+8-1=-5, ∴当 a=1 时,曲线 y=f(x)在点(2,f(2))处的切线方程为 5x+y-8=0. (2)f(x)=-x(x-a)2=-x3+2ax2-a2x, f′(x)=-3x2+4ax-a2=-(3x-a)(x-a), a 令 f′(x)=0,解得 x=3或 x=a. 由于 a≠0,以下分两种情况讨论. ①若 a>0,当 x 变化时,f′(x)的正负如下表: x f ′( x ) a (-∞,3) - a 3 0 a (3,a) + a 0 (a,+∞) -

a a a 4 因此,函数 f(x)在 x=3处取得极小值 f(3),且 f(3)=-27a3. 函数 f(x)在 x=a 处取得极大值 f(a),且 f(a)=0. ②若 a<0,当 x 变化时,f′(x)正负如下表: x (-∞,a) a a (a,3) a 3 a (3,+∞)

f ′( x )



0



0



因此函数 f(x)在 x=a 处取得极小值 f(a),且 f(a)=0; a a a 函数 f(x)在 x=3处取得极大值 f(3),且 f(3)= 4 -27a3. a (3)证明:由 a>3,得到3>1,当 k∈ [-1,0]时,k-cosx≤1,k2-cosx≤1. 由(2)知 f(x)在(-∞,1]上是减函数,要使 f(k-cosx)≥f(k2-cos2x)(x∈R),只 要 k-cosx≤k2-cos2x(x∈R). 即 cos2x-cosx≤k2-k(x∈R).○* 1 1 设 g(x)=cos2x-cosx=(cosx-2)2-4, 则函数 g(x)在 R 上的最大值为 2. 要使○*式恒成立必须 k2-k≥2,即 k≥2 或 k≤-1. 所以,在区间[-1,0]上存在 k=-1,使得 f(k-cosx)≥ f(k2-cos2x)对任意的 x∈R 恒成立. 5.设 A、B 两地相距 30 千米,如下图所示在它们之间铺设一条铁道,由于 地质条件不同,在 y>0 地区铺设费用为 105 元/千米,而在 y≤0 地区铺设费用为 6×104 元/千米,求最经济的铺设线路.

解析 设∠ADx=θ,由图形的对称性可知,研究 y 轴的一侧即可.由点 A 5 向 x 轴作垂线, 设垂足为 C, 则 CD=5cotθ, AD=sinθ.∴DO=15-DC=15-5cotθ. 设铺设铁道的费用为 P,则 5 P=(15-5cotθ)×6×104+sinθ×105. 1 5 cosθ P′=3×105· - 5 × 10 sinθ sin2θ. 3 5cosθ 令 P′=0,得sin2θ= sin2θ . 3 4 3 ∴cosθ=5,此时 sinθ=5,cotθ=4. 45 ∴OD= 4 . 45 45 在 x 轴上取点 D( 4 , 0), E(- 4 , 0), 则 AD→DE→EB 为最佳经济铺设线路.


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