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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 \(g(x) = \frac{x^4}{4} + 10px\).**

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

#### **Step 4: Set \(f'(x) = 0\).**

Since \(e^{g(x)} \neq 0\) for any real \(x\), the equation \(f'(x) = 0\) simplifies to:

\[
x^3 + 10p = 0
\]

#### **Step 5: Solve for \(x\).**

\[
x^3 = -10p
\]

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

---

### **Answer:**

The function \(f(x) = e^{x^4 / 4 + 10px}\) has a critical point at:

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

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Do you have any questions, or would you like further details?  

Here are 5 related questions you might explore:  
1. What is the significance of critical points in function analysis?  
2. How does the chain rule apply to exponential functions?  
3. What if the derivative was undefined at some points—how would that affect critical points?  
4. How would the answer change if \(p\) were a function of \(x\)?  
5. What happens if \(p = 0\)? Where are the critical points then?

**Tip:** Remember, functions involving exponentials often require careful use of the chain rule when differentiating.

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.

Learn how to find the critical points of the function f(x) = e^(x^4 / 4 + 10px) using differentiation techniques like the chain rule. This problem involves the calculation of critical points by finding the derivative and setting it equal to zero. Suitable for advanced high school calculus students in Grades 11-12.

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