Consider the function and its derivatives: $f(x) = x \cdot e^{-3x}$, $f'(x) = (3x - 1) \cdot e^{-3x}$ $f''(x) = 3 \cdot (3x - 2) \cdot e^{-3x}$ a) What is the domain of $f$ in interval notation? b) Find the set of all $x$-values of the $x$-intercepts $f$. c) Pick the correct response: The function $f$ is neither even or odd. d) Enter the set of all $y$-values where $f$ has a horizontal asymptote (Hint: $\lim_{x \to \infty} \frac{x}{e^x} = 0$): e) Enter the set of all $x$-values where $f$ has a vertical asymptote: f) Find the set of all critical numbers of $f$. g) Find the intervals of increase and decrease for $f$. $f$ is increasing on: $f$ is decreasing on: h) Enter the set of all $x$-values where $f$ has a local maximum: i) Enter the set of all $y$-values where $f$ has a local minimum: j) Find the intervals of concavity for $f$. $f$ is concave up on: $f$ is concave down on: k) Enter the set of all $x$-values where $f$ has an inflection point.
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In interval notation, this is $(-\infty, \infty)$. Show more…
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