Question

State whether the following statements are true or false. Give reasons for your answers.
a. the function f:R³→R, givan by f(x,y,z)=|x|+|y|+|z| is differential at (2,3,-1)

Answer 1

Consider $f$ on $S=(0,+\infty)\times(0,+\infty)\times(-\infty,0)$.

We have $f(x,y,z)=x+y-z$ on $S$.

So $f$ is differentiable on $S$. Since $S$ is an open set, and $(2,3,-1)\in S$, we obtain that $f$ is differentiable at $(2,3,-1)$

Answer: $f$ is differentiable at $(2,3,-1)$