The position of a moving car at time

Question:

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

  1. (1) $\left(t_{2}-t_{1}\right) / 2$

  2. (2) $\hat{a}\left(t_{2}-t_{1}\right)+b$

  3. (3) $\left(t_{1}+t_{2}\right) / 2$

  4. (4) $2 a\left(t_{1}+t_{2}\right)+b$


Correct Option: , 3

Solution:

Average speed $=f^{\prime}(t)=\frac{f\left(t_{2}\right)-f\left(t_{1}\right)}{t_{2}-t_{1}}$

$2 a t+b=a\left(t_{1}+t_{2}\right)+b \Rightarrow t=\frac{t_{1}+t_{2}}{2}$

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