Key Concepts and Formulas

• Average Speed: The average speed over an interval $[t_1, t_2]$ is given by $V_{avg} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$.
• Instantaneous Speed: The instantaneous speed at time $t$ is given by the derivative of the position function: $V_{inst}(t) = f'(t)$.
• Mean Value Theorem (MVT): If $f(t)$ is continuous on $[t_1, t_2]$ and differentiable on $(t_1, t_2)$, there exists a $t \in (t_1, t_2)$ such that $f'(t) = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$.

Step-by-Step Solution

Step 1: Understand the Given Information

We are given the position function of a car: $f(t) = at^2 + bt + c$, where $a, b, c > 1$ and $t > 0$. We want to find the time $t$ in the interval $[t_1, t_2]$ where the instantaneous speed equals the average speed.

Step 2: Calculate the Average Speed

The average speed over the interval $[t_1, t_2]$ is calculated as follows: $V_{avg} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$ Substitute the position function: $V_{avg} = \frac{(at_2^2 + bt_2 + c) - (at_1^2 + bt_1 + c)}{t_2 - t_1}$ Simplify: $V_{avg} = \frac{a(t_2^2 - t_1^2) + b(t_2 - t_1)}{t_2 - t_1}$ Factor the difference of squares: $V_{avg} = \frac{a(t_2 - t_1)(t_2 + t_1) + b(t_2 - t_1)}{t_2 - t_1}$ Factor out $(t_2 - t_1)$: $V_{avg} = \frac{(t_2 - t_1)[a(t_2 + t_1) + b]}{t_2 - t_1}$ Cancel $(t_2 - t_1)$, since $t_1 \neq t_2$: $V_{avg} = a(t_1 + t_2) + b$

Step 3: Calculate the Instantaneous Speed

The instantaneous speed is the derivative of the position function with respect to time: $V_{inst}(t) = f'(t) = \frac{d}{dt}(at^2 + bt + c)$ Using the power rule, we get: $V_{inst}(t) = 2at + b$

Step 4: Equate Average Speed and Instantaneous Speed

We want to find the time $t$ when $V_{avg} = V_{inst}(t)$: $a(t_1 + t_2) + b = 2at + b$ Subtract $b$ from both sides: $a(t_1 + t_2) = 2at$ Divide by $2a$, since $a > 1$ and thus $a \neq 0$: $t = \frac{a(t_1 + t_2)}{2a} = \frac{t_1 + t_2}{2}$

Step 5: State the Conclusion

The average speed is attained at $t = \frac{t_1 + t_2}{2}$. This is the midpoint of the interval $[t_1, t_2]$.

Common Mistakes & Tips

• Mean Value Theorem: Recognizing this as an MVT problem simplifies the approach.
• Algebraic Manipulation: Careful simplification of the average speed expression is crucial.
• Quadratic Shortcut: For quadratic position functions, the time where instantaneous velocity equals average velocity is always the midpoint of the interval.

Summary

We found the average speed of the car over the interval $[t_1, t_2]$ and the instantaneous speed at time $t$. By equating these two expressions and solving for $t$, we found that the average speed is attained at $t = \frac{t_1 + t_2}{2}$.

The final answer is \boxed{{{\left( {{t_1} + {t_2}} \right)} \over 2}}, which corresponds to option (A).