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The problem asks to find the value of \( x \) for which the function \( f(x) = e^{x^4/4 + 10px} \) has a critical point, and the answer should include \( p \).

### Step 1: Find the first derivative of \( f(x) \)
To locate critical points, we need to take the derivative of \( f(x) \) and set it equal to zero. First, differentiate the function \( f(x) \).

\[
f(x) = e^{x^4/4 + 10px}
\]

Using the chain rule, the derivative is:

\[
f'(x) = e^{x^4/4 + 10px} \cdot \frac{d}{dx}\left(\frac{x^4}{4} + 10px\right)
\]

Now differentiate the exponent \( \frac{x^4}{4} + 10px \):

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

Thus, the first derivative becomes:

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

### Step 2: Set the derivative equal to zero
To find the critical points, set \( f'(x) = 0 \):

\[
e^{x^4/4 + 10px} \cdot (x^3 + 10p) = 0
\]

Since the exponential function is never zero, we have:

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

### Step 3: Solve for \( x \)
Solve the equation \( x^3 + 10p = 0 \) for \( x \):

\[
x^3 = -10p
\]

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

### Final Answer:
The value of \( x \) for which \( f(x) \) has a critical point is:

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

Would you like more details or have any questions?

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Here are 5 follow-up questions related to this problem:

1. What is the significance of a critical point in the context of a function?
2. How would you check if the critical point found is a maximum, minimum, or saddle point?
3. How does the value of \( p \) affect the location of the critical point?
4. What happens to the critical point when \( p = 0 \)?
5. How would the second derivative of \( f(x) \) help in analyzing the nature of the critical point?

**Tip:** To find whether the critical point is a maximum or minimum, use the second derivative test by checking the sign of \( f''(x) \) at the critical point.

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

Learn how to find the critical point of the function f(x) = e^(x^4/4 + 10px) by taking the derivative and solving for x. This problem uses the first derivative test and involves exponential functions. Suitable for college-level calculus students.

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