The problem asks for the value of $x$ at which the function $f(x) = e^{x^4/4 + 10px}$ has a critical point, where $p$ is a parameter.

Step 1: Find the derivative of $f(x)$

To find the critical points, we first need to take the derivative of $f(x)$. Since $f(x) = e^{g(x)}$, where $g(x) = \frac{x^4}{4} + 10px$, we apply the chain rule.

$f'(x) = \frac{d}{dx}\left( e^{g(x)} \right) = e^{g(x)} \cdot g'(x)$

Next, we calculate $g'(x)$. The derivative of $g(x)$ is:

$g'(x) = \frac{d}{dx} \left( \frac{x^4}{4} + 10px \right) = x^3 + 10p$

Thus, the derivative of $f(x)$ is:

$f'(x) = e^{x^4/4 + 10px} \cdot (x^3 + 10p)$

Step 2: Set the derivative equal to zero

Critical points occur when $f'(x) = 0$. Since $e^{x^4/4 + 10px}$ is never zero (the exponential function is always positive), we set the other factor equal to zero:

$x^3 + 10p = 0$

Step 3: Solve for $x$

Solving for $x$:

$x^3 = -10p$

$x = -\sqrt[3]{10p}$

Thus, the critical point occurs at $x = -\sqrt[3]{10p}$.

Would you like more details or have any further questions on this? Here are some related questions to explore:

1. What are inflection points, and how do you find them for the same function?
2. How does the value of $p$ affect the location of the critical point?
3. How would you classify the critical point as a maximum, minimum, or saddle point?
4. Can there be multiple critical points if we modify the function in a certain way?
5. What would change in the process if the function was $f(x) = e^{x^6/6 + 10px}$?

Tip: Critical points occur where the derivative is zero or undefined. For exponential functions, the factor inside the exponential never becomes zero, so focus on the remaining terms!