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[answered] Make sure you know what it means for a n n matrix to be pos


Could someone who is really good in math help me out with this project and assure me 100 out of 100 or at least 95 with providing me a full answers with steps written in word doc? thank you in advance.


1. Make sure you know what it means for a n ? n matrix to be positive definite, negative definite, and

 

indefinite. Explain this in the write up. There are several equivalent definitions, so mention at least

 

two. Also describe the second derivative test for critical points and their Hessian matrices. 2. Show that the matrix

 

is indefinite by finding different vectors ~v ?R3 such that

 

~vTM~v can be positive sometimes and negative sometimes. (Find them by trial and error.) 3. Show the matrix in # 2 is indefinite by finding its eigenvalues. 4. Show the matrix in #2 is indefinite by computing determinants of the north-west sub-matrices. 5. For the function f(x,y) = sin(x)cos(y):

 

a) Find all critical points in the region [??,?] ? [??,?].

 

b) Compute the Hessian matrix at each critical point.

 

c) Classify each critical point by looking at the eigenvalues of the Hessian matrix. 6. The function f(x,y,z) = x2 +x2y +y2z +z2?4z has a critical point at (0,2,0). Find all the critical points for

 

this function (there are 5 total), and classify them all (if possible) using the Hessian matrix and the

 

second derivative test.

 


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