f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围
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![f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围](/uploads/image/z/5421712-40-2.jpg?t=f%28x%29%3D2x%5E2%2F%28x%2B1%29+g%28x%29%3Dasin%28%CF%80%2F6%2Ax%29-2a%2B2+%28a%3E0%29%E8%8B%A5%E5%AD%98%E5%9C%A8x1%2C2%E2%88%88%5B0%2C1%5D%E4%BD%BF%E5%BE%97f%28x1%29%3Dg%28x2%29%E6%88%90%E7%AB%8B%2C%E6%B1%82%E5%AE%9E%E6%95%B0a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4f%28x%29%3D2x%5E2%2F%28x%2B1%29+g%28x%29%3Dasin%28%CF%80%2F6%2Ax%29-2a%2B2+%28a%3E0%29%E8%8B%A5%E5%AD%98%E5%9C%A8x1%2Cx2%E2%88%88%5B0%2C1%5D%E4%BD%BF%E5%BE%97f%28x1%29%3Dg%28x2%29%E6%88%90%E7%AB%8B%EF%BC%8C%E6%B1%82%E5%AE%9E%E6%95%B0a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围
f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围
f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围
f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围f(x)=2x^2/(x+1) g(x)=asin(π/6*x)-2a+2 (a>0)若存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,求实数a的取值范围
f'(x)=[4x(x+1]-2x^2)/(x+1)^2
=2(x^2+2x)/(x+1)^2
=2(x+1)^2/(x+1)^2-2/(x+1)^2
=2-2/(x+1)^2
2-2/(x+1)^2=0
单调递增,所以f(x)∈[0,1]
x∈[0,1]时
g(x)∈[-2a+2,-3a/2+2] (a>0)
要想有解则:
-3a/2+2≥0
a≤4/3
-2a+2≤1
a≥1/2
则1/2≤a≤4/3
a范围为(1/2,4/3]
f(x)'>0 (x∈[0,1])单调递增,所以f(x)∈[0,1]
x∈[0,1]时g(x)∈[-2a+2,-a+2] (a>0)
要想有解则:-a+2>=0且-2a+2<=1解得1/2<=a<=2
过程我不再写了 。你也着急等。。。
最后是求 两个区间【0,1】和【-2a+2,(-3/2)a+2】有交集
求得满足条件的结果是【2/3,1】。。
f(x)求导后知道在0-1上是增函数,所以值域为0-1使g(x)有解即可,1<=a<= 2/3为空集
存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,
即当x∈[0,1]时,f(x) 和g(x)值域有交集。
f(x)=2x^2/(x+1) =[2(x+1)2-4(x+1)+2)]/(x+1)
= [ 2(x+1)+2/(x+1) ]-4
≥2√ 2...
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存在x1,x2∈[0,1]使得f(x1)=g(x2)成立,
即当x∈[0,1]时,f(x) 和g(x)值域有交集。
f(x)=2x^2/(x+1) =[2(x+1)2-4(x+1)+2)]/(x+1)
= [ 2(x+1)+2/(x+1) ]-4
≥2√ 2(x+1)+2/(x+1) -4 (均值不等式,x=0取等号)
=0
g(x)=asin(π/6*x)-2a+2
-2a+2 ≤ g(x) ≤-3/2a +2
要使f(x) 和g(x)值域有交集,-3/2a +2≥0,即a≤4/3,由a>0
a范围为(0 4/3]
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