Let's solve the given problem step-by-step.

Problem:

For what value of $x$ does the function $f(x) = e^{x^4 / 4 + 10px}$ have a critical point?
Answer with an expression that includes $p$.

Solution:

Step 1: Find the critical points.

Critical points occur where the derivative of the function is zero or undefined. We need to compute the derivative $f'(x)$ with respect to $x$ and set it equal to zero.

Given: $f(x) = e^{x^4 / 4 + 10px}$

We will use the chain rule for differentiation.

Step 2: Differentiate $f(x)$.

Since $f(x) = e^{g(x)}$, where $g(x) = \frac{x^4}{4} + 10px$, the derivative is:

$f'(x) = e^{g(x)} \cdot g'(x)$

Now, we need to compute $g'(x)$.

Step 3: Differentiate $g(x) = \frac{x^4}{4} + 10px$.