To analyze the function $f(x) = \frac{x^2 e^x}{x^2 + e^x}$, we can explore its key properties, such as limits, derivatives, and possible critical points. Here's a breakdown of the function's characteristics:

1. Domain

The function is defined for all $x \in \mathbb{R}$ since both the numerator and the denominator are defined for all real numbers.

2. Limits

• As $x \to -\infty$: f(x) \to 0 \quad \text{(since e^x \to 0$faster than$x^2 grows)}

• As $x \to \infty$: f(x) \to 1 \quad \text{(as both x^2$and$e^x$grow, but$e^x dominates)}

3. Derivative

To find critical points and analyze the behavior of the function, we can compute the derivative $f'(x)$ using the quotient rule: $f'(x) = \frac{(x^2 + e^x)(2xe^x) - (x^2 e^x)(2x + e^x)}{(x^2 + e^x)^2}$ This derivative can be simplified to find critical points where $f'(x) = 0$.

4. Critical Points and Behavior

Solving $f'(x) = 0$ will yield the critical points. We can also check the second derivative to determine concavity and the nature of these critical points.

5. Graphing

Plotting $f(x)$ will provide visual insight into its behavior, including intercepts, asymptotes, and overall shape.

Would you like to explore any specific aspect of this function further, such as finding critical points or sketching the graph?

Additional Questions

1. What is the behavior of the function around its critical points?
2. How do the limits at infinity influence the shape of the graph?
3. Can we find the maximum or minimum values of the function?
4. What happens to the function at specific values like $x = 0$?
5. Would you like to see the graph of the function?

Tip

When analyzing rational functions, checking limits and derivatives can provide critical information about their behavior without needing extensive calculations.