Consider the function f(x)a =x²x ≤ - 3ax + b- 3 < x < 4x³ + 4x - 91x ≥ 4Determine the values of the constants a and b that make f(x) continuous.b =- x - 2ififif

The given function is,The function should be continuous so we can write,

Answered: x²x2 x - 3 ax + b -3 < x < 4 x³ + 4x x…

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

Consider the function f(x)
a =
x²
x ≤ - 3
ax + b
- 3 < x < 4
x³ + 4x - 91
x ≥ 4
Determine the values of the constants a and b that make f(x) continuous.
b =
- x - 2
if
if
if

Expert Solution

Step 1: limit at x=-3

The given function is,

$f open parentheses x close parentheses equals open curly brackets table row cell x squared minus x minus 2 comma thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space x less or equal than negative 3 end cell row cell a x plus b comma thin space thin space thin space thin space thin space thin space thin space thin space thin space minus 3 less than x less than 4 end cell row cell x cubed plus 4 x minus 91 comma thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space x greater or equal than 4 end cell end table close$

The function should be continuous so we can write,

$limit as x rightwards arrow negative 3 to the power of minus of f open parentheses x close parentheses equals limit as x rightwards arrow negative 3 to the power of plus of f open parentheses x close parentheses limit as x rightwards arrow negative 3 to the power of minus of open parentheses x squared minus x minus 2 close parentheses equals limit as x rightwards arrow negative 3 to the power of plus of open parentheses a x plus b close parentheses open parentheses negative 3 close parentheses squared minus open parentheses negative 3 close parentheses minus 2 equals a open parentheses negative 3 close parentheses plus b 9 plus 3 minus 2 equals negative 3 a plus b 10 equals negative 3 a plus b b equals 3 a plus 10 space............. left parenthesis 1 right parenthesis$