Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 48


$ x=0$, local minima at $ x=0$.

Work Step by Step

To find the critical point, we put $ f'(x)=0$, so we have $$ f(x)=e^{-x}+x\Longrightarrow f'(x)=-e^{-x}+1=0,$$ then $ e^{-x}=1$ and hence $ x=0$. So the critical point is $ x=0$. Moreover, we have $$ f''(x)=e^{-x}\Longrightarrow f''(0)=1>0$$ then $ f(x) $ has local minima at $ x=0$.
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