For what value of a is the following piecewise function differentiable everywhere? f(x) = = Sax² – 4, 3a x − 2x, a = -1 a = 2 a = 0 a = -2 x < 1 x > 1 For no value of a, f(x) will be differentiable at x = 1.
For what value of a is the following piecewise function differentiable everywhere? f(x) = = Sax² – 4, 3a x − 2x, a = -1 a = 2 a = 0 a = -2 x < 1 x > 1 For no value of a, f(x) will be differentiable at x = 1.
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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![**Piecewise Function Differentiability Problem**
**Question:**
For what value of \( \alpha \) is the following piecewise function differentiable everywhere?
\[
f(x) =
\begin{cases}
a x^2 - 4, & x \leq 1 \\
3ax - 2x, & x > 1
\end{cases}
\]
**Options:**
1. \( \alpha = -1 \) (This option is selected)
2. \( \alpha = 2 \)
3. \( \alpha = 0 \)
4. \( \alpha = -2 \)
5. For no value of \( \alpha \), \( f(x) \) will be differentiable at \( x = 1 \)
This question requires the determination of the value of \( \alpha \) that makes the piecewise function \( f(x) \) differentiable at the point \( x = 1 \). Differentiability at this point implies that the left-hand derivative and the right-hand derivative at \( x = 1 \) must be equal. Moreover, the function should also be continuous at \( x = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c3310c5-cf1b-4860-895e-b0d1f5af2b55%2Fbb8d1bf9-b791-4bca-90d1-ff6c4580d5de%2F6u0ltop_processed.png&w=3840&q=75)
Transcribed Image Text:**Piecewise Function Differentiability Problem**
**Question:**
For what value of \( \alpha \) is the following piecewise function differentiable everywhere?
\[
f(x) =
\begin{cases}
a x^2 - 4, & x \leq 1 \\
3ax - 2x, & x > 1
\end{cases}
\]
**Options:**
1. \( \alpha = -1 \) (This option is selected)
2. \( \alpha = 2 \)
3. \( \alpha = 0 \)
4. \( \alpha = -2 \)
5. For no value of \( \alpha \), \( f(x) \) will be differentiable at \( x = 1 \)
This question requires the determination of the value of \( \alpha \) that makes the piecewise function \( f(x) \) differentiable at the point \( x = 1 \). Differentiability at this point implies that the left-hand derivative and the right-hand derivative at \( x = 1 \) must be equal. Moreover, the function should also be continuous at \( x = 1 \).
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