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We learned this gradient vector stuff in class but after much struggling I just got the hang of it. I am not very flexible with it. All i know is that at a point the vector that points to the direction with highest rate of change is the gradient vector. I have a bit of confusion here. I found the gradient vector by squaring Fx and Fy then square rooting them to get gradient vector at point (x,y). However I found gradient vector at all those 4 points (4 quadrants) to be equal to sqrt2/e^2 which is strange to me. How can I tell from gradient vector if they are max or min? Using wolfram I got quadrant 2,4 are mins and 1,3 are max but I dont know how gradient tells whether the extrema are max or min.

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First, what you found to be $\sqrt2/\mathrm e^2$ is the norm of the gradient vector, not the gradient vector. Second, the exercise is very badly phrased; a quadrant doesn't "represent" a local maximum or minimum, it contains one. Third, you can't directly find minima and maxima from individual gradient vectors, but in the present case the idea is to use the level curves to see that there are extrema near $(\pm0.7,\pm0.7)$ and then use the gradients at $(\pm1,\pm1)$ to decide which of those extrema are minima or maxima by finding whether the gradient points towards or away from the extremum.

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