Question

3. This problem is to prove the following in the precise fashion described in class: Let o sR be open and let f :o, R have continuous partial derivatives of order three. If (o, 3o) ▽f(zo. ) = (0,0),Jar( , ) < 0, and fzz(z ,m)f (zo,yo) -(fe (a ,yo)) a local maximum value at (zo, yo) (that is, there exists r 0 such that B,(zo, yo) S O and f(a, y) 3 f(zo, yo) for all (x, y) e Br (o, o) is partial derivatives of order three. If (xo, yo) O, 2 > 0, then f achieves A number of steps for this problem have been worked out in class. These steps are given below You are expected to include proofs of these steps even though proofs were given in class. Here are the steps gve dinvahon Multivariable Taylor Polynomial: Suppose that f has continuous partial derivatives of order thre on an open set containing (oo)(,) and the segment joining (2o,30) and (z,y). Then there is a point P on the segment joining (ro, vo) and (x,y) such that 2! + R(x,y) where (b) Let a,b,c be real constants, and let ф(x,y) az? + 2bzy + cy2. If a < 0 and ac-b2 > 0, then ф(z, y) 0 for all (zw) E R2. If a > 0 and ac-b2 > 0, then ф(z,0)2 0 for all (z,y) e R2. If 0 and ac-b2 < 0, then along one line φ(x,y) > 0 except at (0,0) where φ(0,0)-0 and along another line d(x,y) < 0 except at (0,0). (e) This is a no brainer. If (, y) E R2 is fixed and a, b,c are real numbers (possibly depending on z and y) for which a < 0 and ac ->0, then az2+2bay+ cy' s0 As noted in in class, you should be able to put (a) and (c) together to prove the given statement.

Answer 1

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