3e6a3f…: “Without the navy fish”

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Suppose that $F(x,y) =x^3+3xy$ . Then, $\grad {F}(x,y) = \vector {\answer {3x^2+3y},\answer {3x}}.$

How many distinct points $(x,y)$ are there for which $\grad {F}(x,y) = \vec {0}$ ?

none one two more than two, but finitely many infinitely many

If $\grad {F}(x,y) = \vec {0}$ , we must have that the $x$ and $y$ components are zero simultaneously. Note that the only way the $y$ -component can be zero occurs when $x=0$ . How many options does this leave for the $x$ -component?

The point $(x,y)$ for which $\grad {F}(x,y) = \vec {0}$ is $(x,y) = \left (\answer {0},\answer {0} \right )$

As it turns out, the points for which $\grad {F}(x,y) = \vec {0}$ will be very important when we try to find relative extrema of a function of two variables. This exercise gives a little practice with analyzing the gradient to that end.