# Question: Let be a continuous real valued

Let be a continuous real-valued function on the unit circle {x € R2 : |x| =1} such that

f (0, 1) = g(1, 0) = 0 and g(- x )= - g(x).

Define f: R2→R by

F (x) = {|x| . g (x/|x| x ≠0,

0 x = 0.

(a) If x € R2 and h: R →R is defined by h (t) = f (tx) show that h is differentiable.

(b). show that f is not differentiable at (0, 0) unless g = 0.

f (0, 1) = g(1, 0) = 0 and g(- x )= - g(x).

Define f: R2→R by

F (x) = {|x| . g (x/|x| x ≠0,

0 x = 0.

(a) If x € R2 and h: R →R is defined by h (t) = f (tx) show that h is differentiable.

(b). show that f is not differentiable at (0, 0) unless g = 0.

## Answer to relevant Questions

Let f : R2 →R be defined by f(x, y) = { x|y| / √x2 + y2 (x, y) ≠ 0, 0 (x, y) =0.Use the theorems of this section to find f1 for the following: a. f(x, y, z) = xy b. f(x, y) = sin (xsin (y)). c. f(x, y, z) = sin (xsin (ysin (z)) d. f(x, y, z) = xy2 e. f(x, y, z) =xy+z f. f(x, y, z) =(x + ...Find the partial derivatives of the following functions: a. f(x,y,z)=xy b. f(x,y,z)=z c. f(x,y)=sin (xsin (y)) d. f(x,y,z)= sin (x sin (y sin(z))) e. f(x,y,z)=xy2 f. f(x,y,z)=xy=z g. f(x,y,z)=(x +y)2 h. f(x,y)= ...Define g, h: {x€R2} |x| ≤ 1} →R by g(x,y) = (x,y, √1-x2-y2), h(x,y) = (x,y, - √1-x2-y2),Let A C Rn be an open set and f : A→ Rn a continuously differentiable 1-1 function such that det f1 (x) ≠ 0 for all . Show that f (A) is an open set and f -1: f (A) →A is differentiable. Show also that f ...Post your question