Taylor’sTheorem and Applications
1. Find the second degree Taylor polynomial P2(x)=f(a)+∇f(a)6(x−a)+(x−a)TH(a)(x−a)
for the following:
a.f(x,y)=x3+3xy +y3;a=(−1,−1)
b. g(x,y,z)=x2yz +cos(xy);a=(1,π/3,3)
2. Find all critical points for each given function and for each critical point a,
(i).evaluate the Hessian matrix H(a),
(ii).find the eigenvalues of H(a),and
(iii).determine if the xTH(a)xis positive definite,negative definite,or indefinite,and
(iv).classify aas arelative minimum,relative maximum,or saddle point.
.
a.f(x,y)=x3+3xy +y3
b. g(x,y)=3x2y+y3−3x2−3y2+2
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