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The function given is:

\[
f(x) = e^{1 - \sqrt{x^2}} = e^{1 - |x|}
\]

### To find the possible maximum and minimum points, we follow these steps:

1. **Take the derivative of \( f(x) \):**
   The first derivative helps us find the critical points, which may correspond to maxima, minima, or saddle points.

\[
f'(x) = \frac{d}{dx} \left( e^{1 - |x|} \right)
\]

Using the chain rule, the derivative of \( e^{1 - |x|} \) involves differentiating the exponent. For \( x \geq 0 \), \( |x| = x \), and for \( x < 0 \), \( |x| = -x \). Therefore, we treat the cases separately.

For \( x > 0 \):
\[
f'(x) = e^{1 - x} \cdot (-1) = -e^{1 - x}
\]

For \( x < 0 \):
\[
f'(x) = e^{1 + x} \cdot (-1) = -e^{1 + x}
\]

At \( x = 0 \), \( f'(x) \) is continuous because \( f(x) = e^{1} \), and the absolute value function is non-differentiable at \( x = 0 \), but the first derivative from both sides approaches the same value.

2. **Find the critical points:**
   Set \( f'(x) = 0 \).

Since \( e^{1 - |x|} \) is never zero (the exponential function is always positive), the derivative does not give any zeroes. Therefore, there are no critical points where \( f'(x) = 0 \).

However, we check the boundary behavior and analyze the function's behavior as \( x \to \infty \) and \( x \to -\infty \).

3. **Analyze the function at boundary points:**

- As \( x \to \infty \), \( f(x) = e^{1 - |x|} \to 0 \).
- As \( x \to -\infty \), \( f(x) = e^{1 - |x|} \to 0 \).
- At \( x = 0 \), \( f(0) = e^{1} \), which gives the highest value of the function.

Thus, the function has a **global maximum** at \( x = 0 \), where \( f(0) = e^1 \), and the function tends to 0 as \( x \to \infty \) or \( x \to -\infty \), indicating that it has no other critical points or local minima.

### Conclusion:
- **Global Maximum**: At \( x = 0 \), \( f(x) = e^1 \).
- The function tends to 0 as \( |x| \to \infty \), with no local minima.

Would you like further details or clarifications?

Here are 5 related questions for further exploration:
1. How would the behavior of the function change if the exponent were altered to \( e^{1 - x^2} \)?
2. What happens to the derivative and critical points if the function involves higher powers of \( x \)?
3. Can we classify whether the maximum found is an absolute or relative maximum?
4. How does the absolute value function \( |x| \) affect the continuity and differentiability of functions?
5. What is the significance of boundary behavior when evaluating limits for \( x \to \infty \)?

**Tip**: When taking derivatives of functions involving absolute values, always split the function into different cases based on the sign of the variable.

Find the possible maximum and minimum points for f(x) = e^(1 - sqrt(x^2))

This problem explores finding the possible maximum and minimum points for the function f(x) = e^(1 - sqrt(x^2)). Using derivatives and critical point analysis, we determine that the function has a global maximum at x = 0 and approaches zero as |x| tends to infinity.

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