Question

经过幂函数y=x^(1/3)和幂函数y=x^a(a为常数）图像的公共点分别作切线,若两数线的夹角是45度,求a.

Answer 1

x^(1/3)=x^a,x=1或x=0.对y=x^(1/3)求导y`=(1/3)x^(-2/3),代入x=1,k1=1/3,

对y=x^(a)求导,y`=ax^(a-1),代入x=1,k2=a,

若两线的夹角是45度,tan45=(k1-k2)/(1+k1*k2),a=1/2,

或tan45=(k2-k1)/(1+k1*k2),a=2