To find information about the function $y = 7t^2 + 6e^t$, we could perform several analyses, such as finding its derivative, second derivative, or critical points. Here are a few possible directions:

1. First Derivative (dy/dt): To determine the rate of change of $y$ with respect to $t$, we take the derivative: $\frac{dy}{dt} = \frac{d}{dt}(7t^2) + \frac{d}{dt}(6e^t)$ Applying the power rule and exponential rule: $\frac{dy}{dt} = 14t + 6e^t$

2. Second Derivative (d²y/dt²): To analyze the concavity or the acceleration of change, we find the second derivative: $\frac{d^2y}{dt^2} = \frac{d}{dt}(14t) + \frac{d}{dt}(6e^t)$ $\frac{d^2y}{dt^2} = 14 + 6e^t$

3. Critical Points: Set the first derivative equal to zero to find critical points: $14t + 6e^t = 0$ This equation could be solved numerically for specific values of $t$.

4. Behavior Analysis:

   • As $t \to \infty$, the term $6e^t$ dominates, so $y \to \infty$.
   • As $t \to -\infty$, $6e^t$ approaches zero, so $y \approx 7t^2$, which still tends to infinity.

5. Graphical Analysis: Plotting the function $y = 7t^2 + 6e^t$ can give insight into the behavior, concavity, and growth rate of the function.

Would you like to explore any of these points in more detail?

---

Expanding Knowledge

1. What are the critical points of the function?
2. How does the function behave as $t$ approaches zero?
3. What is the significance of the second derivative in understanding concavity?
4. How would the function change if we used a different coefficient for $t^2$ or $e^t$?
5. How does the exponential term $e^t$ affect the long-term behavior of the function compared to polynomial growth?

Tip: Analyzing the first and second derivatives of a function gives essential insights into its growth and shape, especially when combining polynomial and exponential terms.