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Find all points from the domain of $$ f(x,y)=e^{x^2-xy-1} $$ in which the function f reaches the maximum rate change (I mean gain/increase) in the direction of x-axis

the domain: (am I right?) $$ x \in R, y \in R $$ so the domain is real numbers>0.

I thought about directional derivative and it's property: $$ [e^{x^2-xy-1}(2x-y), e^{x^2-xy-1}(-x)]*[cos\alpha, cos\beta] $$ where $$ cos\alpha=1, cos\beta=0 $$ so it's: $$ e^{x^2-xy-1}(2x-y) $$ and what should I do now?

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2 Answers 2

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You now have a function (call this g(x,y)) that gives the rate of change of f in the direction of the x-axis for every point (x,y). The question remains when is g(x,y) a maximum? i.e. when is the rate of change of f in the direction of the x-axis a maximum?

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In general you should take the derivative and set it equal to zero to find extrema, to find out if it is a (local) minimum or maximum you should take a look at the second derivative.

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