In higher dimensions, exactly the same statement holds; however, the proof is slightly more complicated. The complication is that in 1 dimension, one can either move left or right from a point, while in higher dimensions, one can move in many directions. Thus, if the derivative does not vanish, one must argue that there is some direction in which the function increases – and thus in the opposite direction the function decreases. This is the only change to the proof or the analysis.
