Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 4 - Applications of the Derivative - 4.1 Maxima and Minima - 4.1 Exercises - Page 243: 61

Answer

(a) $(0, 0)$ and $(3, \frac{27}{e^3})$ are the critical points. (b) The absolute maximum of $f$ on the given interval is about 1.344, and the absolute minimum is $ −e ≈ −2.718 $.

Work Step by Step

(a) $f'(x) = 3x^2e^{−x}+x^3 · (−e^{−x}) = e^{−x} · x^2 · (3 − x)$. This expression is zero when $x = 0$ and when $x = 3$, so $(0, 0)$ and $(3, \frac{27}{e^3})$ are the critical points. (b) $f(−1) = −e$, $f(0) = 0$, $f(3) = \frac{27}{e^3} ≈ 1.344$, and $f(5) ≈ .8422$. So the absolute maximum of $f$ on the given interval is about 1.344, and the absolute minimum is $ −e ≈ −2.718 $.
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