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For what value of x does the function  f (x) have a critical point if

 f (x)  =  e x4/4 + 10px 

Answer with an expression that includes p.'

To find the critical points of the function \( f(x) = e^{\frac{x^4}{4}} + 10px \), we need to locate where its derivative \( f'(x) \) is zero or undefined.

### Step 1: Differentiate \( f(x) \) with respect to \( x \)
Using the chain rule, the derivative \( f'(x) \) is:
\[
f'(x) = \frac{d}{dx}\left( e^{\frac{x^4}{4}} \right) + \frac{d}{dx}(10px)
\]

1. For the first term, \( \frac{d}{dx}\left( e^{\frac{x^4}{4}} \right) = e^{\frac{x^4}{4}} \cdot \frac{d}{dx} \left( \frac{x^4}{4} \right) = e^{\frac{x^4}{4}} \cdot x^3 \).
   
2. The derivative of \( 10px \) is simply \( 10p \), since \( p \) is treated as a constant.

So,
\[
f'(x) = e^{\frac{x^4}{4}} \cdot x^3 + 10p
\]

### Step 2: Set \( f'(x) = 0 \)
To find the critical points, set \( f'(x) = 0 \):
\[
e^{\frac{x^4}{4}} \cdot x^3 + 10p = 0
\]

Since \( e^{\frac{x^4}{4}} \) is always positive for real values of \( x \), we can divide by it without changing the equation:
\[
x^3 + 10p = 0
\]

Solving for \( x \):
\[
x^3 = -10p
\]
\[
x = -\sqrt[3]{10p}
\]

### Answer
The critical point occurs at:
\[
x = -\sqrt[3]{10p}
\]

Would you like further details on this process or have additional questions?

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Here are five related questions to expand your understanding:

1. How would the critical point change if \( p \) were negative?
2. What happens to the critical point if \( p = 0 \)?
3. Are there any conditions under which this function \( f(x) \) would have no critical points?
4. How can you determine if the critical point is a minimum or maximum?
5. What is the effect of changing the exponent in the term \( x^4/4 \)?

**Tip:** Remember that a critical point occurs where the derivative equals zero or is undefined—key information for analyzing function behavior.

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

Explore how to find the critical points of the function f(x) = e^(x^4/4) + 10px. This detailed solution demonstrates calculus concepts including derivatives, exponentials, and the chain rule. Suitable for advanced high school students in Grades 11-12.

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