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While it is simple to grasp limits in multiple variables, since the formal definition extends in the obvious way, I am having a harder time grasping the same concept with critical points and extreme values. I am asking for an answer deeply based in the single-variable case that extends the problem of finding critical points and extreme values to multiple variables. In particular, an explanation emphasizing the sudden importance of vectors (gradients etc.) would be appreciated.

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Good intuition provide geographical maps with contour lines: if you consider the altitude to be a function of $(x,y)$, then those contour lines (or "level curves") are just sets "$f(x,y)=const$". The extreme points can be a "top of a hill", or "bottom of a lake/crater" or saddle (if you are in a saddle, then in one direction the ridge goes up and in another direction you go down the valley, but in all directions the first derivative of the altitude is zero, similar to the $\pm x^2$ function at $0$).

There are also more complicated critical points but these are the basic ones; the defining condition is that the "tangent" plane in a critical point is completely horizontal. In general, gradient is the direction of the "steepest slope" in the "up-hill" direction and it is zero in critical points.

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