https://file.aiquickdraw.com/mathbot/file/172909872392903jilzc0.jpg

The problem asks for the value of \( x \) at which the function \( f(x) = e^{x^4 / 4 + 10px} \) has a critical point. Here's how we can approach it:

### Steps to Solve:
1. **Find the derivative** of \( f(x) \). A critical point occurs where the first derivative \( f'(x) \) equals zero or does not exist.

   - The function is \( f(x) = e^{g(x)} \), where \( g(x) = \frac{x^4}{4} + 10px \).
   - Use the chain rule for differentiation:
     \[
     f'(x) = e^{g(x)} \cdot g'(x)
     \]
   - Now, differentiate \( g(x) \):
     \[
     g'(x) = \frac{d}{dx}\left( \frac{x^4}{4} + 10px \right) = x^3 + 10p
     \]
   - Therefore, the derivative of \( f(x) \) is:
     \[
     f'(x) = e^{\left(\frac{x^4}{4} + 10px\right)} \cdot (x^3 + 10p)
     \]

2. **Set the derivative equal to zero** to find the critical points:
   \[
   e^{\left(\frac{x^4}{4} + 10px\right)} \cdot (x^3 + 10p) = 0
   \]
   Since \( e^{\left(\frac{x^4}{4} + 10px\right)} \) is never zero (because the exponential function is always positive), we can simplify this to:
   \[
   x^3 + 10p = 0
   \]
   Solving for \( x \):
   \[
   x^3 = -10p \quad \Rightarrow \quad x = -\sqrt[3]{10p}
   \]

### Conclusion:
The critical point occurs at \( x = -\sqrt[3]{10p} \).

Would you like a deeper explanation of any step, or do you have further questions?

### Related Questions:
1. What is the second derivative of \( f(x) \) at the critical point?
2. How do you classify the critical point as a maximum, minimum, or saddle point?
3. What happens to the critical point when \( p = 0 \)?
4. Can the function have more than one critical point if we modify \( f(x) \)?
5. How does the parameter \( p \) affect the location of the critical point?

### Tip:
When solving for critical points in exponential functions, the exponential factor can be disregarded when finding where the derivative is zero because the exponential function never equals zero.

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.

Explore how to find the critical point of the function f(x) = e^(x^4/4 + 10px) using calculus. This problem involves finding the derivative, applying the chain rule, and solving for x. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation