a08863…: “Ahead of a proud fact”

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The following exercise explores the nature of finding critical points. You may assume that each function below is differentiable for all $(x,y) \in \R ^2$ . Suppose that $F(x,y)$ is a function, and it is known that $\grad {F}(x,y) = \vector {x^2+2x,y^2-4y}$ . Then $F(x,y)$ has $\answer {4}$ distinct critical point(s).

The $x$ -values for the critical points are $x=\answer {-2}$ and $x=\answer {0}$ (type the smaller $x$ -value first) and the $y$ -values for the critical points are $y=\answer {0}$ and $y= \answer {4}$ (type the smaller $y$ -value first).

Select all of the following that are critical points.

$(-2,0)$ $(-2,4)$ $(0,0)$ $(0,4)$

Suppose that $G(x,y)$ is a function, and it is known that $\grad {G}(x,y) = \vector {x+2y,y^2+2x}$ . Then $G(x,y)$ has $\answer {2}$ distinct critical point(s).

The $x$ -values for the critical points are $x=\answer {-8}$ and $x=\answer {0}$ (type the smaller $x$ -value first) and the $y$ -values for the critical points are $y=\answer {0}$ and $y= \answer {4}$ (type the smaller $y$ -value first).

Select all of the following that are critical points.

$(-8,0)$ $(-8,4)$ $(0,0)$ $(0,4)$

Suppose that $H(x,y)$ is a function, and it is known that $\grad {H}(x,y) = \vector {x+2y,2x-y-1}$ . Then $H(x,y)$ has $\answer {1}$ distinct critical point(s).

The critical point is $\left (\answer {\frac {2}{5}}, \answer {-\frac {1}{5}} \right )$ .

After working through each of these parts, can you understand what makes the number of critical points these functions have different from each other?