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Question: multivariable calculusadvanced math using lagrangian equation to solve this problem...

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multi-variable calculus(advanced math)

using Lagrangian equation to solve this problem with the information (theorem provided below......) and need full proof, thank you.

Question 3attached below.

Question 4. Make the same assumptions as in Question 3. a. Using Question 3 and Theorem 1, prove that if v . ▽F(a) = 0, then ▽φ(a) . v-0. b. Using part (a) and linear algebra, prove that (6) is true. For example, one linear algebra fact we remember is that given a nonzero vector w and an arbitrary vector z, we can always decompose z in the form z-aw + z, where a e R and orthogonal to w. You may also remember more advanced aspects of linear algebra is

In the next two questions, you will use Theorem 1 to prove the Lagrange Multiplier rule for constrained minimization problems, which here we will write in the form minimize φ(x) F(x) = 0 subject to the constraint where F are C1 functions Rn R. We assume that a is a point at which the above problem is solved. In other words, we assume that F(a)-0 and if F(x)-0, then φ(a) 6(x). We also assume that VF(a) 0. Our goal is to prove that there exists μ E R such that Vo(a) = ,IVF(a).

Theorem 1. Assume that F:R- R is Cl, let S denote the zero locus of F (see (1), and assume that a is a point in S such that ▽F(a) 0. If v ERı is a vector such that v·▽F(a) = 0, then v is tangent to F at a in the sense of Definition 1 Here we have assumed that VF(a) 0. Clearly, this holds if and only if д¡F(a) 0 for some J E {1, , n). For the proof, we will assume that this holds for j n, in other words that an F(a) f 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.

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