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.