Intuitively, a differentiable function is approximated by its derivative – a differentiable function behaves infinitesimally like a linear function formula_23 or more precisely, formula_24 Thus, from the perspective that "if "f" is differentiable and has non-vanishing derivative at formula_25 then it does not attain an extremum at formula_25" the intuition is that if the derivative at formula_13 is positive, the function is "increasing" near formula_25 while if the derivative is negative, the function is "decreasing" near formula_29 In both cases, it cannot attain a maximum or minimum, because its value is changing. It can only attain a maximum or minimum if it "stops" – if the derivative vanishes (or if it is not differentiable, or if one runs into the boundary and cannot continue). However, making "behaves like a linear function" precise requires careful analytic proof.