Show that f ( x , y ) = x 2 y e − x 2 − y 2 has maximum values at ( ± 1 , 1 / 2 ) and minimum values at ( ± 1 , − 1 / 2 ) . Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points?
Show that f ( x , y ) = x 2 y e − x 2 − y 2 has maximum values at ( ± 1 , 1 / 2 ) and minimum values at ( ± 1 , − 1 / 2 ) . Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points?
Show that
f
(
x
,
y
)
=
x
2
y
e
−
x
2
−
y
2
has maximum values at
(
±
1
,
1
/
2
)
and minimum values at
(
±
1
,
−
1
/
2
)
. Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points?
Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
2. Find a matrix A with the following qualities
a. A is 3 x 3.
b. The matrix A is not lower triangular and is not upper triangular.
c. At least one value in each row is not a 1, 2,-1, -2, or 0
d. A is invertible.
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