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 :

• A. $a \left( t _{2}- t _{1}\right)+ b$
• B. $\left(t_{2}-t_{1}\right) / 2$
• C. $2 a\left(t_{1}+t_{2}\right)+b$
• D. $\left(t_{1}+t_{2}\right) / 2$

Answer 1

Answer: (D)

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