Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Finding Horizontal Tangents of a Function**
**Question:**
For which values of \( x \) does \( f(x) = 3e^x - 6x \) have a horizontal tangent line?
**Solution Approach:**
1. **Find the derivative \( f'(x) \):**
To determine where the function has a horizontal tangent line, we need to find its derivative and set it equal to zero. The function provided is:
\[
f(x) = 3e^x - 6x
\]
Using the rules of differentiation, we find:
\[
f'(x) = 3e^x - 6
\]
2. **Set the derivative equal to zero and solve for \( x \):**
\[
f'(x) = 0 \implies 3e^x - 6 = 0
\]
Solving this equation for \( x \):
\[
3e^x = 6
\]
\[
e^x = 2
\]
\[
x = \ln(2)
\]
Thus, the function \( f(x) = 3e^x - 6x \) has a horizontal tangent line at \( x = \ln(2) \).
**Diagram Explanation:**
The diagram depicted on the paper shows annotations and a logical flow of solving the derivative and setting it to zero. There are arrows pointing out critical steps, such as finding the derivative and indicating where it equals zero, connected to the final solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d5280c1-0888-4121-94fb-2a368f5b3705%2F30186f6e-e36d-4623-a2de-da1efd947cea%2Fe57sj3k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding Horizontal Tangents of a Function**
**Question:**
For which values of \( x \) does \( f(x) = 3e^x - 6x \) have a horizontal tangent line?
**Solution Approach:**
1. **Find the derivative \( f'(x) \):**
To determine where the function has a horizontal tangent line, we need to find its derivative and set it equal to zero. The function provided is:
\[
f(x) = 3e^x - 6x
\]
Using the rules of differentiation, we find:
\[
f'(x) = 3e^x - 6
\]
2. **Set the derivative equal to zero and solve for \( x \):**
\[
f'(x) = 0 \implies 3e^x - 6 = 0
\]
Solving this equation for \( x \):
\[
3e^x = 6
\]
\[
e^x = 2
\]
\[
x = \ln(2)
\]
Thus, the function \( f(x) = 3e^x - 6x \) has a horizontal tangent line at \( x = \ln(2) \).
**Diagram Explanation:**
The diagram depicted on the paper shows annotations and a logical flow of solving the derivative and setting it to zero. There are arrows pointing out critical steps, such as finding the derivative and indicating where it equals zero, connected to the final solution.
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