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Find a and b so that the function s both continuous and differentiable. a = f(x) = = 5x³ - 3x² + 4, x < −2, { ax+b, x ≥ −2

Find a and b so that the function s both continuous and differentiable. a = f(x) = = 5x³ - 3x² + 4, x < −2, { ax+b, x ≥ −2

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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Find a and b so that the function is both continuous and differentiable. a = b = -50 √ 5x³ − 3x² +4, x < −2, ax + b, x > -2 f(x) = {
Transcribed Image Text:Find a and b so that the function is both continuous and differentiable. a = b = -50 √ 5x³ − 3x² +4, x < −2, ax + b, x > -2 f(x) = {
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the given function is 

f\left(x\right)=\:\left\{\begin{matrix}5x^3-3x^2+4,\:\:\:\:\:\:\:x<-2\\ ax+b,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:x\ge -2\end{matrix}\right.

the function is continuous at x=-2 so the left-hand side and right-hand limits of the function at x=-2 should be equal

\lim _{x\to -2^-}f\left(x\right)=\lim _{x\to -2^+}f\left(x\right)

\lim _{x\to -2^-}\left(5x^3-3x^2+4\right)=\lim _{x\to -2^+}\left(ax+b\right)

5\left(-2\right)^3-3\left(-2\right)^2+4=a\left(-2\right)+b

-40-12+4=-2a+b

-48=-2a+b

b=2a-48

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