Real Analysis IIPlease follow exact hint and steps 8.s.) Complete the Implicit Function Theorem for F: R¹ R2where F(x) = (F1(x), F2(x)) and x = (x1, x2, x3, x4):Let F R¹ R2 beF(=(0, 0), and(F1, F₂)(x3, x4)where F = (F₁, F₂), c = (C₁, C2, C3, C4) € R¹,-(c) = detThen there exists a neighborhood N of (c₁, c₂) and C¹ functions→ R.03:04:such that x3 = 03(1, 2) and 4 = $4(1, 2) solve F(x1, x2, 3, 4)(0, 0) for allEN and 03 (C1, C2) = C3 and 4 (C₁, C2) == C4.→RChoices:D4F2(c) 0 D4F1(c) N (x1, x2)D3F1(c) N D3F2(c) C1C

Answered: Image /qna-images/answer/0193c811-bc4d-42c2-9b81-ab0a4e7de604.jpg

8. s.) Complete the Implicit Function Theorem for F: R4 R² where F(x) = (F1(x), F2(x)) and x = = (x1, x2, x3, x4): Let F: R¹ R2 be F = (0,0), and a(F₁, F2) (x3, x4) where F = (F1, F2), c = (C₁, C2, C3, C4) E R4, such that x3 = (0, 0) for all -(c) = det Then there exists a neighborhood N of (C₁, C2) and C¹ functions 03: → R. * →R 03(1, 2) and 4 = 4(x1, x2) solve F(x1, 2, 3, 4) = EN and 03 (C1, C2) = C3 and 04 (C1, C₂) = C4.