Answer to Question #105494 in Calculus for khushi

Question

State whether the following statements are true or false. Give reasons for your answers (1) The function f:R^3®R, given by f( x, y, z) =|x|+|y|+|z| is differentiable at (2, 3,-1).

Answer 1

There exist neighbourhood U of (2, 3, -1) such that$\forall (x, y, z) \in U, f(x, y, z) = x + y - z$ by definition of the absolute value.So, if $(x, y, z) \in U:\\$ $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = 1\\ \frac{\partial f}{\partial z} = -1\\$ Since function $g(x, y, z) = const$ is continuous, all partial derivatives of f(x, y, z) are continuous at (2, 3, -1). Thus, f(x, y, z) is differentiable at (2, 3, -1).