The given problem asks for the value of $x$ where the function $f(x)$ has a critical point. The function is:

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

Steps to find the critical points:

1. Find the derivative of $f(x)$: The critical points occur where the first derivative of $f(x)$ equals zero, i.e., $f'(x) = 0$.

   The derivative of $f(x)$ with respect to $x$ is computed using the chain rule and product rule. We need to differentiate both terms in the function.

   • For the term $e^{x^4 / 4}$, apply the chain rule: $\frac{d}{dx} \left( e^{x^4 / 4} \right) = e^{x^4 / 4} \cdot \frac{d}{dx} \left( \frac{x^4}{4} \right) = e^{x^4 / 4} \cdot x^3$
   • The derivative of $10px$ is simply $10p$, since $p$ is treated as a constant.

   Thus, the first derivative of $f(x)$ is: $f'(x) = e^{x^4 / 4} \cdot x^3 + 10p$

2. Set the derivative equal to zero to find the critical points: $e^{x^4 / 4} \cdot x^3 + 10p = 0$

3. Solve for $x$: First, subtract $10p$ from both sides: $e^{x^4 / 4} \cdot x^3 = -10p$

   Since $e^{x^4 / 4}$ is always positive for all real values of $x$, it cannot be zero or negative. Thus, the sign of $x^3$ must be such that it can balance the negative $-10p$ term.

   The solution is $x^3 = \frac{-10p}{e^{x^4 / 4}}$.

   Unfortunately, solving this equation algebraically requires numerical methods or further approximation techniques depending on the context of the problem, as the expression is implicit.

Would you like further details on solving this equation, or do you have any additional questions?

Five related questions:

1. What are the conditions for a point to be classified as a critical point?
2. How do you classify critical points as maxima, minima, or saddle points?
3. Can this function have multiple critical points? If so, under what conditions?
4. How does the value of $p$ affect the position of the critical points?
5. Can you apply numerical methods to approximate the solution to $x^3 = \frac{-10p}{e^{x^4 / 4}}$?

Tip:

When working with implicit equations involving exponentials and polynomials, it's often useful to consider numerical methods or graphical approaches to approximate solutions.