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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i need step 3 please
![**Find the critical numbers of the function.**
Given:
\[ h(t) = t^{3/4} - 4t^{1/4} \]
---
**Step 1**
For \( h(t) = t^{3/4} - 4t^{1/4} \), we have:
\[
h'(t) = \frac{3}{4}t^{-\frac{1}{4}} - \frac{1}{4}(3t^{-3/4})
\]
Which simplifies to:
\[
h'(t) = \frac{3\sqrt{t} - 4}{4\sqrt[4]{t^3}}
\]
---
**Step 2**
Critical numbers occur where \( h'(t) = 0 \) and where \( h'(t) \) is undefined.
Set:
\[
0 = h'(t) = \frac{1}{4}t^{-3/4}(3t^{1/2} - 4) = \frac{3t^{1/2} - 4}{4\sqrt[4]{t^3}}
\]
Solve for when the numerator is 0:
- This occurs at \( t = \frac{16}{9} \).
Your answer is \( \frac{16}{9} \).
---
**Step 3**
The expression \( h'(t) = \frac{3t^{1/2} - 4}{4\sqrt[4]{t^3}} \) is undefined when the denominator is 0. This occurs where \( t \) equals the following values:
- The incorrect solution provided was \( t = \frac{16}{9} \) which doesn't satisfy the undefined condition.
Ensure your answers are entered as a comma-separated list.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7f2a813-5af6-4b89-aa65-b7da82bb3773%2F240a4322-436f-449c-8a6f-2391caaa5e8b%2Fextynr3_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the critical numbers of the function.**
Given:
\[ h(t) = t^{3/4} - 4t^{1/4} \]
---
**Step 1**
For \( h(t) = t^{3/4} - 4t^{1/4} \), we have:
\[
h'(t) = \frac{3}{4}t^{-\frac{1}{4}} - \frac{1}{4}(3t^{-3/4})
\]
Which simplifies to:
\[
h'(t) = \frac{3\sqrt{t} - 4}{4\sqrt[4]{t^3}}
\]
---
**Step 2**
Critical numbers occur where \( h'(t) = 0 \) and where \( h'(t) \) is undefined.
Set:
\[
0 = h'(t) = \frac{1}{4}t^{-3/4}(3t^{1/2} - 4) = \frac{3t^{1/2} - 4}{4\sqrt[4]{t^3}}
\]
Solve for when the numerator is 0:
- This occurs at \( t = \frac{16}{9} \).
Your answer is \( \frac{16}{9} \).
---
**Step 3**
The expression \( h'(t) = \frac{3t^{1/2} - 4}{4\sqrt[4]{t^3}} \) is undefined when the denominator is 0. This occurs where \( t \) equals the following values:
- The incorrect solution provided was \( t = \frac{16}{9} \) which doesn't satisfy the undefined condition.
Ensure your answers are entered as a comma-separated list.
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