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I'm given the graph shown here: enter image description here

And the question is to determine the signs of the partial derivatives (a) fx (1,2) and (b) fy(1,2)

My thought was for (a), as I hold y constant and move positive on x, then what is the value of z. I'm not sure how I could really see that from this graph. At the point of x=1 the z graph looks flat....book says positive.

Then on (b), I was thinking hold x constant and move positive on the y axis...looks like z is neg there, which is what the book has.

Is there a better way to look at this? If I'm holding one of the variables constant, what do I do with the y value in (a) and with the x value in (b)?

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First, $x=1$ does not designate a point on the graph, the point on the graph is $(1,2,f(1,2))$ and the graph is not flat there.

Holding the value of $y\,$ fixed at a value of $2$ but increasing $x$ beyond a value of $1$ results in an increase in the value of $z$.

Holding the value of $x$ fixed at a value of $1$ but increasing $y$ beyond a value of $2$ results in a decrease in the value of $z$.

Thus $f_x(1,2)>0$ and $f_y(1,2)<0$.

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