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Find a and b such that the function f(x) = {x² - 6x +10 [x² - 6x +10 if x≤-1 if x>-1 Is differentiable everywhere a b=

Find a and b such that the function f(x) = {x² - 6x +10 [x² - 6x +10 if x≤-1 if x>-1 Is differentiable everywhere a b=

Calculus: Early Transcendentals
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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**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 \( b \):** - \( a = \) - \( b = \)
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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