已知函数f(x)=10+2x-x2,g(x)=f(2-x2),则函数g(x)的单调区间是?
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![已知函数f(x)=10+2x-x2,g(x)=f(2-x2),则函数g(x)的单调区间是?](/uploads/image/z/8621652-12-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D10%2B2x-x2%2Cg%28x%29%3Df%282-x2%29%2C%E5%88%99%E5%87%BD%E6%95%B0g%28x%29%E7%9A%84%E5%8D%95%E8%B0%83%E5%8C%BA%E9%97%B4%E6%98%AF%3F)
已知函数f(x)=10+2x-x2,g(x)=f(2-x2),则函数g(x)的单调区间是?
已知函数f(x)=10+2x-x2,g(x)=f(2-x2),则函数g(x)的单调区间是?
已知函数f(x)=10+2x-x2,g(x)=f(2-x2),则函数g(x)的单调区间是?
f(x)=10+2x-x^2,g(x)=f(2-x^2),这是复合函数
g(x)= 10+2(2-x^2) - (2-x^2)^2 = -x^4 + 2 x^2 +10
g(x)= -(x^2 -1)^2 +11 ,当 x^2 1 时,g(x) 单调递减.
综上,当 -1
g(x)=f(2-x2)=10+2(2-x2)-(2-x2)2
=-x^4+2x^2+10
g(x)'=-4x^3+4x
令g(x)'=-4x^3+4x=0
则x1=0,x2=1,x3=-1
x<-1或0<x<1时,g(x)'<0,函数单调递减
-1<x<0或x>1时,g(x)'>0,函数单调递增
f(x)=10+2x-x^2
g(x)
=f(2-x^2)
=10+2(2-x^2)-(2-x^2)^2
=10+4-2x^2-x^4-4+4x^2
=10+2x^2-x^4
g`(x)=4x-4x^3>0
x-x^3>0
x(1-x^2)>0
x(x^2-1)<0
x<-1 0
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f(x)=10+2x-x^2
g(x)
=f(2-x^2)
=10+2(2-x^2)-(2-x^2)^2
=10+4-2x^2-x^4-4+4x^2
=10+2x^2-x^4
g`(x)=4x-4x^3>0
x-x^3>0
x(1-x^2)>0
x(x^2-1)<0
x<-1 0
x(x^2-1)>0
-1
单调增区间 x<-1 0
收起