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.