# The position of a moving car at time t is given

Question:

The position of a moving car at time $t$ is given by $f(\mathrm{t})=\mathrm{at}^{2}+\mathrm{bt}+\mathrm{c}, \mathrm{t}>0$, where $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are real numbers greater than 1 . Then the average speed of the car over the time interval $\left[\mathrm{t}_{1}, \mathrm{t}_{2}\right]$ is attained at the point :

1. $\mathrm{a}\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right)+\mathrm{b}$

2. $\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right) / 2$

3. $2 \mathrm{a}\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)+\mathrm{b}$

4. $\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right) / 2$

Correct Option: , 4

Solution:

$\frac{f\left(\mathrm{t}_{2}\right)-f\left(\mathrm{t}_{1}\right)}{\mathrm{t}_{2}-\mathrm{t}_{1}}=2 \mathrm{at}+\mathrm{b}$

$\frac{\mathrm{a}\left(\mathrm{t}_{2}^{2}-\mathrm{t}_{1}^{2}\right)+\mathrm{b}\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right)}{\mathrm{t}_{2}-\mathrm{t}_{1}}=2 \mathrm{at}+\mathrm{b}$

$\Rightarrow \mathrm{a}\left(\mathrm{t}_{2}+\mathrm{t}_{1}\right)+\mathrm{b}=2 \mathrm{at}+\mathrm{b}$

$\Rightarrow \mathrm{t}=\frac{\mathrm{t}_{1}+\mathrm{t}_{2}}{2}$