NOTES ON CHAPTER 31. Set-upIn what follows, f : R2→ R is continuous function, a = (x0, y0) ∈ R2is a fixed point in R2.2. First Taylor formulaIf f is differentiable at a, there exists a function ω1(x) : R2→ R (depending on a and f) such that(f(x) = f(a) + ∇f(a) · (x − a) + kx − ak ω1(x)limx→aω1(x) = 0This is an immediate consequence of the fact that f is differentiable at a, which is the statement thatlimx→af(x)−f(a)−∇f(a)·(x−a)kx−ak= 0.2.1. Corollary: first order approximation. For x = (x, y) near a = (x0, y0),f(x) ≈ f(a) + ∇f (a) · (x − a)in other wordsf(x, y) ≈ f (x0, y0) +∂f∂x(x0, y0)(x − x0) +∂f∂y(x0, y0)(y − y0)3. Second Taylor formulaIf f is a C2function (it has second order partial derivatives and these are continuous) then there existsa (second order error) function ω2(x) (depending on f and a) such that(f(x) = f(a) + ∇f (a) · (x − a) +12Hf(a)[x − a] + kx − ak2ω2(x)limx→aω2(x) = 0where:Hf(a) =fxx(a) fxy(a)fy x(a) fy y(a)is the Hessian of f at a, and notice that it is a symmetric matrix. Also, the square-bracket action isgiven byHf(a)[x − a] = Hf(a)(x − a) · (x − a)Note: The proof of the second Taylor formula is not as s traight-forward as the one for the first Taylorformula. Consult the textbook for details.3.1. Corollary: second order approximation. Let x = (x, y) near a = (x0, y0). Unravelling the terminvolving the HessianHf(a)[x − a] = [x − x0, y − y0] ·fxx(a) fxy(a)fy x(a) fy y(a)·x − x0y − y0= fxx(a)(x − x0)2+ 2fxy(a)(x − x0)(y − y0) + fy y(a)(y − y0)2we obtain the second order approximation near a = (x0, y0):f(x, y) ≈ f (a) + ∇f(a) · (x − a) +12Hf(a)[x − a]= f(a) + fx(a)(x − x0) + fy(a)(y − y0)+12fxx(a)(x − x0)2+ 2fxy(a)(x − x0)(y − y0) + fy y(a)(y − y0)24. Extrema of real-valued functions4.1. Critical points. Definition a is a critical point for the function f : R2→ R if ∇f (a) = 0(f0(a) = 0).12 NOTES ON CHAPTER 34.2. Local extrema. a ∈ R2is a local maximum of f if there exists a positive number r > 0 such that:for x ∈ Br(a), f(x) ≤ f(a)Similarly for local minimum.Theorem. If a is a local extremum for f, then a is a critical point.Consequence: when searching for local extrema we need to restrict our search to critical points.4.3. Classification of critical points. Ass ume a is critical , ∇f (a) = 0.If det Hf(a) > 0 and fxx(a) > 0 then a is a local minimum for f(1)If det Hf(a) > 0 and fxx(a) < 0 then a is a local maximum for f(2)If det Hf(a) < 0 then a is a saddle point for f(3)If det Hf(a) = 0, a is a degenerate critical point (inconclusive