1. Quiz 2
(1) Let F : U → R be a function defined on an open subset U of R2 such that Fxx, Fxy and Fyy exist and continuous on U. Let (x0, y0) ∈ U such that Fy(x0, y0) 6= 0. By implicit function theorem, there exists a rectangle (x0−α, x0+α)×(y0−β, y0+β) and a differentiable function f : (x0− α, x0+ α) → (y0− β, y0+ β) such that y0 = f (x0) and F (x, f (x)) = 0 for any x ∈ (x0− α, x0+ α). Prove that
f00(x) = FxxFy2− 2FxyFxFy+ FyyFx2
Fy3 .
(2) Find the extremum of F (x, y) = x3+3x2y subject to the condition x2+4xy +5y2 = 5 using following steps.
(a) Let G(x, y) = x2+ 4xy + 5y2. Find ∇G(x, y). Here ∇G(x, y) = (Gx, Gy).
(b) Find ∇F (x, y).
(c) Suppose λ is a real number such that ∇F (x, y) = λ∇G(x, y). Solve for the system equations
∇F (x, y) = λ∇G(x, y)
G(x, y) = 5.
(3) If the ellipse x2 a2 + y2
b2 = 1, (a, b > 0,) is to enclose the circle x2 + y2 = 2y, what values of a, b minimize the area of the ellipse?
1
![In other words, if R(z) is a rational function, then there exist polynomial P (z) and Q(z) in C[z] with Q(z) 6= 0 such that R(z](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)