# The position of moving car at time t is given as $f(t) = 3t^2+bt+c, t>0$ where a , b,c are real no.>1 Then the avg speed of the car over the time interval $[t_1,t_2]$ is attained at the pt. ....   Option: 1 Option: 2 Option: 3 Option: 4

Using mean value theorem

$f'(c)=\frac{f(b)-f(a)}{b-a}$

$f'(t)=\frac{f(t_2)-f(t_1)}{t_2-t_1}$

$\\\Rightarrow2 a t+b=\frac{f\left(t_{2}\right)-f\left(t_{1}\right)}{t_{2}-t_{1}} \\\\\Rightarrow 2 a t+b=\frac{a\left(t_{2}^{2}-t_{1}^{2}\right)+b\left(t_{2}-t_{1}\right)}{t_{2}-t_{1}} \\\\ \Rightarrow 2 a t+b=a\left(t_{2}+t_{1}\right)+b \\\\ \Rightarrow \qquad \;\;t=\frac{t_{1}+t_{2}}{2}$

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