Question

设f(x)=3x^2-x+m,g(x)=lnx,当曲线y=f(x)与y=g(x)有公切线时,求实数m的取值范围

Answer 1

f'(x)=6x-1,

g'(x)=1/x, x>0.

6x-1=f'(x)=g'(x)=1/x, x>0.

0=6x^2-x-1=(2x-1)(3x+1), x=1/2.

ln(1/2)=g(1/2)=f(1/2)=3(1/2)^2-(1/2)+m,

m=1/2 - 3/4 - ln(2) = -1/4 - ln(2)