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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![**Problem:**
Show that there are no points of the graph of \( y = \log_5 x \) where the tangent line is horizontal.
**Solution:**
To determine where the tangent line to the graph of \( y = \log_5 x \) is horizontal, we need to find the derivative \( y' \) and set it to zero.
1. **Find the Derivative:**
The derivative of \( y = \log_5 x \) with respect to \( x \) is:
\[
y' = \frac{d}{dx}(\log_5 x) = \frac{1}{x \ln 5}
\]
2. **Set the Derivative to Zero:**
For the tangent line to be horizontal, the derivative must be zero:
\[
\frac{1}{x \ln 5} = 0
\]
However, \( \frac{1}{x \ln 5} \) is never zero for any real value of \( x \). Thus, there are no points where the derivative is zero and therefore no points where the tangent line is horizontal.
**Conclusion:**
The graph of \( y = \log_5 x \) has no points where the tangent line is horizontal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15976126-8979-495c-a2c3-d70684fdeb4f%2Fc689a7ab-960e-48bb-8ee1-c00fd59eda27%2Fj8xobw_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Show that there are no points of the graph of \( y = \log_5 x \) where the tangent line is horizontal.
**Solution:**
To determine where the tangent line to the graph of \( y = \log_5 x \) is horizontal, we need to find the derivative \( y' \) and set it to zero.
1. **Find the Derivative:**
The derivative of \( y = \log_5 x \) with respect to \( x \) is:
\[
y' = \frac{d}{dx}(\log_5 x) = \frac{1}{x \ln 5}
\]
2. **Set the Derivative to Zero:**
For the tangent line to be horizontal, the derivative must be zero:
\[
\frac{1}{x \ln 5} = 0
\]
However, \( \frac{1}{x \ln 5} \) is never zero for any real value of \( x \). Thus, there are no points where the derivative is zero and therefore no points where the tangent line is horizontal.
**Conclusion:**
The graph of \( y = \log_5 x \) has no points where the tangent line is horizontal.
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