96 CHAPTER 2 As a motivation for the convenience of the differential calculus you could try to calculate the derivative of the modulus function directly from the definition of the derivative. Warning. It is possible that all partial derivatives f (x) , 1 k n, x k exist, but that f : U n (x, ) R is not differentiable at x, as the following example shows. Example 2.2.3 Investigate the differentiability properties at the origin of the function f of two variables defined by f (x) = x 1 x 2 /(x 2 + x 2 ) if x = 0 2 and f (0 2 ) = 0. 1 2 Solution. The function f is not continuous at 0 2 , and so certainly f not differentiable. However, the partial derivative x 1 (0 2 ) equals by definition the derivative of the function g of one variable given by g(x 1 ) = f (x 1 , 0). One has g(x 1 ) = 0, both for x 1 = 0 and for x 1 = 0. f f Therefore, x 1 (0 2 ) = 0. In the same way, x 2 (0 2 ) = 0. Conclusion. The approximation definition of the derivative of a function of one variable can be extended to the case of two or more variables (provided you take the right point of view). The resulting derivative is a vector, not a number. The calculation of this derivative proceeds by calculating the partial derivatives. 2.3 MAIN RESULT: FERMAT THEOREM FOR TWO OR MORE VARIABLES Let x be a point of R n and f : U n (x, ) R a function of n variables, defined on an open ball with center x and radius > 0. ¯ The function f : U (x, ) R might also be defined outside this open ¯ ball, but this is not relevant for the present purpose. Definition 2.3 Main problem: unconstrained n-variable opti- mization. The problem f (x) extr, (P 2.1 ) is called an unconstrained n-variable or n-dimensional problem. Definition 2.4 Local extrema. One says that x is a (point of ) local minimum (maximum) of a function f : U n (x, ) R if there exists a ¯ sufficiently small positive number < such that x is a point of global ¯ minimum (maximum) of f : U n (x, ) R.