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 $.
