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517 HW9 1. Let G : R2 → R be class C 1 , such that G(a, b) = 0 and D2 G(a, b) 6= 0. Prove1 that there exists a continuous real-valued function f , defined on an open interval around a, such that f (a) = b and G(x, f (x)) ≡ 0. Hint: Consider the sequence fn+1 (x) = fn (x) − G(x,fn (x)) D2 G(a,b) . 2. Let A be a symmetric n × n matrix and define q : Rn → R by q(x) = xt Ax. Suppose the restriction of q to the unit sphere {x : |x| = 1} attains a maximum or minimum at the point v. Show that Av = λv for some λ ∈ R. 3. Find the minimum value of f (x) = n−1 (x1 + . . . + xn ) on the surface x1 x2 . . . xn = c (c > 0 constant). Use this to deduce the geometric-arithmetic mean inequality: for positive reals a1 , . . . , an , (a1 . . . an )1/n ≤ n−1 (a1 + . . . + an ). 4. A set P ⊂ Rn is called an (n − 1)-patch if for some i ∈ {1, . . . , n}, there exists a real-valued differentiable function h defined on an open set U ⊂ Rn−1 such that P = {x ∈ Rn : πi (x) ∈ U and xi = h(πi (x))}, where πi (x) := (x1 , . . . , x̂i , . . . , xn ). A set M ⊂ Rn is called an (n − 1)-manifold if for each x ∈ M there is an open set V ⊂ Rn containing x such that V ∩ M is a (n − 1)-patch. Suppose g : Rn → R is class C 1 . Show that {x ∈ Rn : g(x) = 0, g 0 (x) 6= 0} is an (n − 1)manifold. 1 Don’t assume the implicit function theorem here! 1