https://file.aiquickdraw.com/mathbot/file/1729055015758rrre6zg3.jpg

We are tasked with finding the critical number of the function \( f(x) = (8x + 2)e^{-3x} \).

### Steps to solve:

1. **Find the first derivative \( f'(x) \):**
   We need to apply the product rule since the function is a product of two parts: 
   - \( u(x) = (8x + 2) \)
   - \( v(x) = e^{-3x} \)
   
   Using the product rule:
   \[
   f'(x) = u'(x)v(x) + u(x)v'(x)
   \]

   First, calculate the derivatives of \( u(x) \) and \( v(x) \):
   - \( u'(x) = 8 \)
   - \( v(x) = e^{-3x} \), so \( v'(x) = -3e^{-3x} \)

   Now apply the product rule:
   \[
   f'(x) = (8)e^{-3x} + (8x + 2)(-3e^{-3x})
   \]
   \[
   f'(x) = 8e^{-3x} - 3(8x + 2)e^{-3x}
   \]

2. **Factor the expression:**
   We can factor out \( e^{-3x} \) from both terms:
   \[
   f'(x) = e^{-3x} \left[ 8 - 3(8x + 2) \right]
   \]
   Simplify inside the brackets:
   \[
   f'(x) = e^{-3x} \left[ 8 - 24x - 6 \right]
   \]
   \[
   f'(x) = e^{-3x} \left[ 2 - 24x \right]
   \]
   \[
   f'(x) = e^{-3x} (2 - 24x)
   \]

3. **Find critical points:**
   Critical points occur where \( f'(x) = 0 \). Since \( e^{-3x} \) is never zero, we only need to solve:
   \[
   2 - 24x = 0
   \]
   \[
   24x = 2
   \]
   \[
   x = \frac{2}{24} = \frac{1}{12}
   \]

Thus, the critical number is \( x = \frac{1}{12} \).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can we verify that \( x = \frac{1}{12} \) is the only critical point?
2. How would you determine if this critical point is a maximum or minimum?
3. What is the second derivative of the function?
4. How would you apply the product rule to different types of functions?
5. What does the exponential factor \( e^{-3x} \) contribute to the shape of the graph?

**Tip:** Always check the second derivative or use the first derivative test to determine the nature of critical points.

The function f(x) = (8x + 2)e^{-3x} has one critical number. Find it.

Learn how to find the critical number of the function f(x) = (8x + 2)e^{-3x}. This problem involves calculus concepts such as derivatives, the product rule, and identifying critical points. A step-by-step solution suitable for students in Grades 11-12 or Calculus I.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation