Solution Found!
Non differentiability? Consider the following functions
Chapter 12, Problem 43E(choose chapter or problem)
Nondifferentiability? Consider the following functions f.
(a) Is f continuous at (0,0)?
(b) Is f differentiable at (0,0)?
(c) If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\).
(d) Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0).
(e) Explain why Theorems 12.5 and 12.6 are consistent with the results in part (a)-(d).
\(f(x, y)= \begin{cases}-\frac{x y}{x^{2}+y^{2}}\) if \((x, y) \neq(0,0)\) \\ 0 & if \((x, y)=(0,0)\end{cases}\)
Questions & Answers
QUESTION:
Nondifferentiability? Consider the following functions f.
(a) Is f continuous at (0,0)?
(b) Is f differentiable at (0,0)?
(c) If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\).
(d) Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0).
(e) Explain why Theorems 12.5 and 12.6 are consistent with the results in part (a)-(d).
\(f(x, y)= \begin{cases}-\frac{x y}{x^{2}+y^{2}}\) if \((x, y) \neq(0,0)\) \\ 0 & if \((x, y)=(0,0)\end{cases}\)
ANSWER:Solution 43E
Step 1:
Given that
Suppose the function f has partial derivatives fx and fy defined on an ope set containing (a, b), with fx and fy continuous at (a, b). Then f is differentiable at (a, b).
If a function f is differentiable at (a, b), then it is continuous at (a, b).