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Question: i need to solve question 1 please write it in...

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I need to solve question 1.

Please write it in clear and readable handwriting. Thank you!!!

Preliminaries In Section 2.4 of the Online Notes, we considered sets S of the form where F : Rn → R is a C1 function. There we gave the following definition: Definition 1: a tangent ขector to S at a point. Given a point a E S such that ▽F(a) 0, a vector v is tangent to S at a if there exists an interval I C R containing 0, and a C function g : I -> R such that g(t) S for all t є 1, g(0) = a, and g (0)-v. In Example 4 in Section 2.4 of the Online Notes, we used the Chain Rule to prove that if v is tangent to S at a, then v·▽F(a) = 0. In this homework, you will use the Implicit Function Theorem to prove the converse of (3), and and you will use that converse to establish the validity of the Lagrange Multiplier Rule, Theorem 1 in Section 2.8 of the Online Notes The converse of (3) is: Theorem 1. Assume that F : Rn → R is C1, let S denote the zero locus of F (see (1)), and assume that a is a point in S such that VF(a) 0 If v ER is a vector such that v VF(a)-0, then v is tangent to F at a in the sense of Definition 1. Here we have assumed that ▽F(a) 0. Clearly, this holds if and only if ỘjF(a)メ0 for some j {1,... ,n}. For the proof, we will assume that this holds for j-n, in other words that anF(a) 0. This is just for notational convenience-we will need the Implicit Function Theorem, and this assumption lets us write it in a convenient way. The general case can be addressed using similar ideas

Warm-up. Assume that F and S are as in Theorem 1 and that 0nF(a)メ0. In this context, the Implicit Function Theorem says that there exist open sets U C Rn-1 and V C R and a C1 function f :U- V such that (αι , . . . , , an-1) EU, an e V, an for x E U × V F (x) 0 if and only if xnf(x1,... ,^n-1) Remind yourself of the formula for o,f(a, derivatives of F. You will need this below. an-1), for j = 1, , n-1, in terms of For both Questions 1 and 2, we assume v is a vector such that We will consider v to be arbitrary but fixed, and we will use the same v for both Questions 1 and 2. Question 1. For v fixed above, satisfying (4), let g(t) = (x1(t),…,xn(t)) be a curve such that Tn We take the Domain(g) to be an interval I C R such that 0 E I and (x1(t),... , xn-i(t) E U whenever t I. You should not write out a proof, but please make sure you understand why such an interval exists Explain how to define xn(t) to guarantee that g(t) E S for all t E 1, g(0) a Your answer should both give a definition of xn(t) and prove that it has the above proper- ties Question 2. For function g(t) whose nth component you found in Question 1, prove that g(0) (This will complete the proof of Theorem 1 in the case when 0nF(a) the idea in general case is basically identical.) 0. As remarked above.

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