Real Analysis IIPlease solve 1b,c&d 1. Let f: R² → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 724 44²if (x, y) = (0,0).(a) Prove that Duf(0, 0) = 0 if u = (1,0).(b) Prove that Duf (0,0) = 0 if u = (0, 1).(c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab ‡0.(d) Prove that Df(0, 0) does not exist (this can be done by provingthat f is not continuous at (0, 0)).1) Fix 4³0. chrose 8 = [E] Assume Q = ||| < 8. Then || = [F(t,0) - f(0,0)] - 0 ||-11/12-07-011=0 < 2₁₁city=(1₁0)L₁: (0,0) (0) = 0 Lufe - 0 | (0,0) 01) Fix 2-0. Choose S-1. Assume 0= 111 < S. Then 11 = [f(0, 1) - ((0,0)] - 011+ 1/[0] = 011=0 < E c+t₁ = (0₁1)L: (0,0)(1)-0 Lake 0/(0,0) = 0Sikunt: ((t,0)=+ (0) = ²00((0,¹) 07:0f =1+ 1²2)si catus (at, bt) f(at, b1) = _abt45²¥² (₁²-b²) ₁²-6₂²=LEH COM

Given: Let f:ℝ2→ℝ where fx,y=0 if…

xy + Let f: R2 → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 72²² if (x, y) = (0,0). (a) Prove that Duf (0,0) = 0 if u = (1,0). (b) Prove that Duf(0, 0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0. (d) Prove that Df(0, 0) does not exist (this can be done by proving that f is not continuous at (0, 0)).

xy + Let f: R2 → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 72²² if (x, y) = (0,0). (a) Prove that Duf (0,0) = 0 if u = (1,0). (b) Prove that Duf(0, 0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0. (d) Prove that Df(0, 0) does not exist (this can be done by proving that f is not continuous at (0, 0)).

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter14: Discrete Dynamical Systems

Section14.3: Determining Stability

Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...

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