来自陶振武的问题
已知函数f(x)=x^3/(x-1)^2,试求f(x)的凹凸区间和拐点急用
已知函数f(x)=x^3/(x-1)^2,试求f(x)的凹凸区间和拐点
急用
1回答
2020-04-28 02:39
已知函数f(x)=x^3/(x-1)^2,试求f(x)的凹凸区间和拐点急用
已知函数f(x)=x^3/(x-1)^2,试求f(x)的凹凸区间和拐点
急用
f(x)=x^3/(x-1)^2
f'(x)={(x-1)^2*3x^2-x^3*2(x-1)}/(x-1)^4
={(x-1)*3x^2-2x^3}/(x-1)^3
=x^2{3x-3-2x}/(x-1)^3
=(x^3-3x^2)/(x-1)^3
f''(x)={(x-1)^3*3x(x-2)-x^2(x-3)*3(x-1)^2}/(x-1)^6
={(x-1)*3x(x-2)-x^2(x-3)*3}/(x-1)^4
=3x{(x-1)(x-2)-x(x-3)}/(x-1)^4
=3x{x^2-3x+2-x^2+3x}/(x-1)^4
=6x/(x-1)^4
x<0时,f''(x)<0,为凸区间;x>0时,f''(x)>0,为凹区间
∴凹区间(0,+∞),凸区间(-∞,0),拐点x=0