Non differentiability Consider the following functions

Chapter 12, Problem 44E

(choose chapter or problem)

Get Unlimited 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).

\(f(x, y)= \begin{cases}\frac{2 x y^{2}}{x^{2}+y^{4}}\) & 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).

\(f(x, y)= \begin{cases}\frac{2 x y^{2}}{x^{2}+y^{4}}\) & if \((x, y) \neq(0,0)\) \\ 0 & if \((x, y)=(0,0)\end{cases}\)

ANSWER:

Step 1 of 7

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).

 

Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back