High school math. Question 2 1, question 3.

It is proved that the tangent points are M(t, t3-t) and N(a, b).

It is easy to know KMN=K tangent, so? =3t2- 1，

It can be changed to 2t3-3at2+a+b=0, ①

Therefore, if the intersection point N(a, b) can be regarded as three tangents of the curve y=f(x),

Then equation ① has three different real roots, g(t)=2t3-3at2+a+b,

Then g'(t)= 6t(t-a),

It is easy to know that the maximum value of g(t) is g(0)=a+b, and the minimum value is g(a)=2a3-3a3+a+b=b-f(a).

To sum up, if the intersection N(a, b) can be used as the three tangents of the curve, then? ,

That is -a < b < f (a).