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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Question
![**Objective:** Find values of \( a \) and \( b \) such that the function \( f(x) \) is differentiable everywhere.
**Function Definition:**
\[
f(x) =
\begin{cases}
x^2 - 6x + 10 & \text{if } x \leq -1 \\
ax + b & \text{if } x > -1
\end{cases}
\]
**Requirements:**
- The function must be continuous at \( x = -1 \).
- The function must be differentiable at \( x = -1 \).
**To Solve:**
1. **Continuity at \( x = -1 \):**
Ensure \( \lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) \).
2. **Differentiability at \( x = -1 \):**
Ensure \( \lim_{x \to -1^-} f'(x) = \lim_{x \to -1^+} f'(x) \).
**Solution:**
- **Find values of \( a \) and \( b \):**
- \( a = \)
- \( b = \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ca42388-a0e6-4236-bc5e-61764756f4fd%2F8c0534c4-8083-4f85-ba5f-2eabd034e2b2%2Fhlq9di_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Objective:** Find values of \( a \) and \( b \) such that the function \( f(x) \) is differentiable everywhere.
**Function Definition:**
\[
f(x) =
\begin{cases}
x^2 - 6x + 10 & \text{if } x \leq -1 \\
ax + b & \text{if } x > -1
\end{cases}
\]
**Requirements:**
- The function must be continuous at \( x = -1 \).
- The function must be differentiable at \( x = -1 \).
**To Solve:**
1. **Continuity at \( x = -1 \):**
Ensure \( \lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) \).
2. **Differentiability at \( x = -1 \):**
Ensure \( \lim_{x \to -1^-} f'(x) = \lim_{x \to -1^+} f'(x) \).
**Solution:**
- **Find values of \( a \) and \( b \):**
- \( a = \)
- \( b = \)
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