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Main points of this past exam are: Hessian Matrix, Critical Points, Function, Critical Points, Local Maximum, Local Minimum, Saddle Points
Typology: Exercises
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3 → R. It has critical points ~a 1
= (2, 2 , 1) and ~a 2
= (− 3 , − 1 , −1). Its
Hessian matrix is
Hf =
2 x
2 y 0 x + y
0 y
2 z z
x + y z x
(a) What is
2 f
∂x∂z
(b) Classify the critical points ~a 1 and ~a 2 as local maximum, local minimum, saddle points, or im-
possible to tell with the given information. Justify your answers.
R
(3x + 2y) dA where R is the region in the xy-plane bounded by the graphs of y = 2 and
2 y = x
2 .
3 → R
3 be given by
F (x, y, z) = (xe
y )ˆi + (z sin y)ˆj + (xy ln z)
k.
(a) Calculate div
(b) Calculate curl
