# Find a and b so that the function e-ax, x ≤ 0 f(x) = {₁² + 5x + b,x>0 is differentiable everywhere. To advance in the circuit, find the sum of a and b.
# Find a and b so that the function e-ax, x ≤ 0 f(x) = {₁² + 5x + b,x>0 is differentiable everywhere. To advance in the circuit, find the sum of a and b.
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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![### Differentiable Functions Analysis
**Question:**
**Find \( a \) and \( b \) so that the function:**
\[
f(x) =
\begin{cases}
e^{-ax}, & \text{if } x \leq 0 \\
x^2 + 5x + b, & \text{if } x > 0
\end{cases}
\]
**is differentiable everywhere.**
To advance in the circuit, find the sum of \( a \) and \( b \).
**Answer:**
\[
\frac{\pi^2 + 8\pi}{8\sqrt{2}}
\]
---
**Explanation of Diagram/Graph:**
This function is split into two parts, one for \( x \leq 0 \) and one for \( x > 0 \). To ensure that \( f(x) \) is differentiable everywhere, especially at \( x = 0 \), the function needs to be continuous and have matching derivatives from both sides at this point. The solution involves solving for the constants \( a \) and \( b \) that meet these conditions and then finding their sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2bb9483f-e549-4e9c-b819-8ce0593a1865%2F450feef5-d377-4005-aa5f-2b58556dabac%2Fzlht576_processed.png&w=3840&q=75)
Transcribed Image Text:### Differentiable Functions Analysis
**Question:**
**Find \( a \) and \( b \) so that the function:**
\[
f(x) =
\begin{cases}
e^{-ax}, & \text{if } x \leq 0 \\
x^2 + 5x + b, & \text{if } x > 0
\end{cases}
\]
**is differentiable everywhere.**
To advance in the circuit, find the sum of \( a \) and \( b \).
**Answer:**
\[
\frac{\pi^2 + 8\pi}{8\sqrt{2}}
\]
---
**Explanation of Diagram/Graph:**
This function is split into two parts, one for \( x \leq 0 \) and one for \( x > 0 \). To ensure that \( f(x) \) is differentiable everywhere, especially at \( x = 0 \), the function needs to be continuous and have matching derivatives from both sides at this point. The solution involves solving for the constants \( a \) and \( b \) that meet these conditions and then finding their sum.
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