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

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$a (t_{2} - t_{1}) + b$

$\frac{t_{1} + t_{2}}{2}$

$\frac{t_{2} - t_{1}}{2}$

$2 a (t_{1} + t_{2}) + b$

answer is A.

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Detailed Solution

Average speed = $= \frac{f (t_{2}) - f (t_{1})}{t_{2} - t_{1}}$

Using Lagranges mean value theorem, we obtain

$\begin{array}{l} f^{'} (t) = \frac{f (t_{2}) - f (t_{1})}{t_{2} - t_{1}} for some t \in [t_{1}, t_{2}] . \\ \Rightarrow 2 a t + b = \frac{(a t_{2}^{2} + b t_{2} + c) - (a t_{1}^{2} + b t_{1} + c)}{t_{2} - t_{1}} \\ \Rightarrow 2 a t + b = \frac{a (t_{2}^{2} - t_{1}^{2}) + b (t_{2} - t_{1})}{t_{2} - t_{1}} \\ \Rightarrow 2 a t + b = a (t_{2} + t_{1}) + b \\ \Rightarrow t = \frac{(t_{2} + t_{1})}{2} \end{array}$