Real Analysis. Multi-variable Calculus. Higher Order Derivatives, Partial Derivatives

Exercise 8.6.7: Follow the strategy below to prove the following simple version of the second derivative test for functions defined on R2 (using (x,y) as coordinates): Suppose f: R2 + R is a twice continuously differentiable function with a critical point at the origin, f'(0,0) = 0. If ?f 2 (0,0) > 0 and 22f (0, 2 af (0,0) 2 >0, ?f -(0,0) then f has a (strict) local minimum at (0,0). Use the following technique: First suppose without loss of generality that f(0,0) = 0. Then prove: a) There exists an A EL(R2) such that g=fo A is such that : (0,0) = 0, and (0,0) = ; (0,0) = 1. b) For any e > 0 there exists a d > 0 such that (g(x,y) x2 - y2| 0 and 22f (0, 2 af (0,0) 2 >0, ?f -(0,0) then f has a (strict) local minimum at (0,0). Use the following technique: First suppose without loss of generality that f(0,0) = 0. Then prove: a) There exists an A EL(R2) such that g=fo A is such that : (0,0) = 0, and (0,0) = ; (0,0) = 1. b) For any e > 0 there exists a d > 0 such that (g(x,y) x2 - y2|