Given: position of the moving car at time tis=f(t)=at2+bt+c
So,
vavg=t2−t1f(t2)−f(t1)
⇒vavg=t2−t1a(t22−t12)+b(t2−t1)
⇒vavg=a(t1+t2)+b
The instantaneous speed is given by:
f′(t)=2at+b
So, to get the point where average speed is equal to car's actual speed,
f′(t)=vavg
⇒a(t1+t2)+b=at+b
⇒t=2t1+t2