Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.1 - Derivatives of Polynomials and Exponential Functions - 3.1 Exercises - Page 182: 71

Answer

Since the derivatives are not equal from the left and the right, the $f$ is not differentiable at $1$. We can see a sketch of $f$ and $f'$ below.

Work Step by Step

$f(x) = x^2+1~~~$ if $x \lt 1$ $f(x) = x+1~~~$ if $x \geq 1$ As $x \to 1^-$, then $~~f'(x) = 2x~~$ and the derivative approaches the value of 2 As $x \to 1^+$, then $~~f'(x) = 1~~$ Since the derivatives are not equal from the left and the right, $f$ is not differentiable at $1$ We can see a sketch of $f$ and $f'$ below.
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