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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!

For what value of x does the function f(x) = e^(x^4/4 + 10px) have a critical point if p is a parameter?

Learn how to find the critical points of the function f(x) = e^(x^4/4 + 10px) using differentiation techniques. This problem explores key calculus concepts like the chain rule, derivatives, and exponential functions. Suitable for undergraduate calculus students.

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