Question

1. Consider the function defined for all real $x$ and $y$ by $f(x, y) = xe^{-y^2 - 4y}$. (a) Find all the critical points of $f$, then classify them by using the second order conditions. (b) Show that $f$ has neither a global maximum nor a global minimum. (c) Let $S = \{(x, y) : 0 \le x \le 5, 0 \le y \le 4\}$. Prove that $f$ has global maximum and minimum points in $S$, then find them. (d) Find the slope of the tangent to the level curve $xe^{-y^2 - 4y} = e^{-4}$ at the point where $x = 1$ and $y = 4 - e$.

          1. Consider the function defined for all real $x$ and $y$ by $f(x, y) = xe^{-y^2 - 4y}$.
(a) Find all the critical points of $f$, then classify them by using the second order conditions.
(b) Show that $f$ has neither a global maximum nor a global minimum.
(c) Let $S = \{(x, y) : 0 \le x \le 5, 0 \le y \le 4\}$. Prove that $f$ has global maximum and minimum points in $S$, then find them.
(d) Find the slope of the tangent to the level curve $xe^{-y^2 - 4y} = e^{-4}$ at the point where $x = 1$ and $y = 4 - e$.

$1. Consider the function defined for all real x and y by f(x, y) = xe^-y^2 - 4y. (a) Find all the critical points of f, then classify them by using the second order conditions. (b) Show that f has neither a global maximum nor a global minimum. (c) Let S = {(x, y) : 0 ≤ x ≤ 5, 0 ≤ y ≤ 4}. Prove that f has global maximum and minimum points in S, then find them. (d) Find the slope of the tangent to the level curve xe^-y^2 - 4y = e^-4 at the point where x = 1 and y = 4 - e.$

Added by Kaitlin S.

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