幂函数 y=x^a(a≠0),当a取不同的正数时,在区间[0,1]上他们的图像是美丽的曲线,设点A(1,0),B(1,0),连接AB,线段AB恰好被这两个幂函数y=x^a,y=x^b图像三等分,既有BM = MN = NA,求a,b的值B(0,1)
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![幂函数 y=x^a(a≠0),当a取不同的正数时,在区间[0,1]上他们的图像是美丽的曲线,设点A(1,0),B(1,0),连接AB,线段AB恰好被这两个幂函数y=x^a,y=x^b图像三等分,既有BM = MN = NA,求a,b的值B(0,1)](/uploads/image/z/12710592-0-2.jpg?t=%E5%B9%82%E5%87%BD%E6%95%B0+y%3Dx%5Ea%28a%E2%89%A00%29%2C%E5%BD%93a%E5%8F%96%E4%B8%8D%E5%90%8C%E7%9A%84%E6%AD%A3%E6%95%B0%E6%97%B6%2C%E5%9C%A8%E5%8C%BA%E9%97%B4%5B0%2C1%5D%E4%B8%8A%E4%BB%96%E4%BB%AC%E7%9A%84%E5%9B%BE%E5%83%8F%E6%98%AF%E7%BE%8E%E4%B8%BD%E7%9A%84%E6%9B%B2%E7%BA%BF%2C%E8%AE%BE%E7%82%B9A%281%2C0%29%2CB%281%2C0%29%2C%E8%BF%9E%E6%8E%A5AB%2C%E7%BA%BF%E6%AE%B5AB%E6%81%B0%E5%A5%BD%E8%A2%AB%E8%BF%99%E4%B8%A4%E4%B8%AA%E5%B9%82%E5%87%BD%E6%95%B0y%3Dx%5Ea%2Cy%3Dx%5Eb%E5%9B%BE%E5%83%8F%E4%B8%89%E7%AD%89%E5%88%86%2C%E6%97%A2%E6%9C%89BM+%3D+MN+%3D+NA%2C%E6%B1%82a%2Cb%E7%9A%84%E5%80%BCB%280%2C1%29)
幂函数 y=x^a(a≠0),当a取不同的正数时,在区间[0,1]上他们的图像是美丽的曲线,设点A(1,0),B(1,0),连接AB,线段AB恰好被这两个幂函数y=x^a,y=x^b图像三等分,既有BM = MN = NA,求a,b的值B(0,1)
幂函数 y=x^a(a≠0),当a取不同的正数时,在区间[0,1]上他们的图像是美丽的曲线,设点A(1,0),B(1,0),连接AB,线段AB恰好被这两个幂函数y=x^a,y=x^b图像三等分,既有BM = MN = NA,求a,b的值
B(0,1)
幂函数 y=x^a(a≠0),当a取不同的正数时,在区间[0,1]上他们的图像是美丽的曲线,设点A(1,0),B(1,0),连接AB,线段AB恰好被这两个幂函数y=x^a,y=x^b图像三等分,既有BM = MN = NA,求a,b的值B(0,1)
三等份
=>
(1,0)/(2/3,1/3)/(1/3,2/3)/(0,1)为端点和两个三分点
(2/3,1/3)满足y=x^a
=>
1/3=(2/3)^a
=>
a=lg(2/3)(1/3)=-lg(2/3)3
(1/3,2/3)满足y=x^b
=>
2/3=(1/3)^b
=>
a=lg(1/3)(2/3)=-lg(3)(2/3)=-(lg(3)2-1)
2 1/2
A(1,0),B(0,1),
∵M,N是三等分点,BM = MN = NA
∴M(1/3,2/3),N(2/3,1/3)
即y=x^a过M(1/3,2/3), ==> 2/3=(1/3)^a ==>a=log(1/3)2/3=1-log3(2)
y=x^b过N(2/3,1/3), ==> 1/3=(2/3)^b ==>b=log(2/3)1/3=1-log(2/3)(2)
A(1,0),B(0,1),AB:y=1-x
∵BM = MN = NA
∴xM=1/3, yM=2/3
xN=2/3,yN=1/3
将M(1/3,2/3)代入y=x^a,
得:2/3=(1/3)^a==>a=log(底1/3)(2/3)
将N(2/3,1/3)代入y=x^b
得:1/3=(2/3)^b==>b= log(底2/3)(1/3)
∴a=log(底1/3)(2/3) b= log(底2/3)(1/3)
A\B怎么是同一个点?