Question

Let f(x,y) = 3x^2 + cos(Pi*y). a) f has a saddle point at (0,k) whenever k is an odd integer b) f has a saddle point at (0,k) whenever k is an even integer) c) f has a local maximum at (0,k) whenever k is an even integer d) f has a local minimum at (0,k) whenever k is an odd integer.

Answer #1

Find the area of one loop of the polar curve r=4*sin(3*theta +
Pi/3)
Let f(x,y) = 3x^2 + cos(Pi*y). a) f has a saddle point at (0,k)
whenever k is an odd integer b) f has a saddle point at (0,k)
whenever k is an even integer) c) f has a local maximum at (0,k)
whenever k is an even integer d) f has a local minimum at (0,k)
whenever k is an odd integer

Let f(x,y) = 3x^2y − 2y^2 − 3x^2 − 8y + 2.
(i) Find the stationary points of f.
(ii) For each stationary point P found in (i), determine whether
f has a local maximum, a local minimum, or a saddle point at P.
Answer:
(i) (0, −2), (2, 1), (−2, 1)
(ii) (0, −2) loc. max, (± 2, 1) saddle points

(3)If H(x, y) = x^2 y^4 + x^4 y^2 + 3x^2 y^2 + 1, show that H(x,
y) ≥ 0 for all (x, y). Hint: find the minimum value of H.
(4) Let f(x, y) = (y − x^2 ) (y − 2x^2 ). Show that the origin
is a critical point for f which is a saddle point, even though on
any line through the origin, f has a local minimum at (0, 0)

Let f(x, y) = (2y-x^2)(y-2x^2) a. Show that f(x, y) has a
stationary point at (0, 0) and calculate the discriminant at this
point. b. Show that along any line through the origin, f(x, y) has
a local minimum at (0, 0)

Find the Critical point(s) of the function f(x, y) = x^2 + y^2 +
xy - 3x - 5. Then determine whether each critical point is a local
maximum, local minimum, or saddle point. Then find the value of the
function at the extreme(s).

Find the location of the critical point of the function
f(x,y)= kx^(2)+3y^(2)-2xy-24y (in terms of k) of
t. The classify the values of k for which the critical
point is a:
I) Saddle Point
II) Local Minimum
III) Local Maximum

Find all local maximum or local minimum or saddle point for f(x,y)=
8y^3 + 12x^2 -24xy

Find the local maximum and minimum values and saddle point(s) of
the function f ( x , y ) = f(x,y)=xe^(-2x2-2y2). If there are no
local maxima or minima or saddle points, enter "DNE."
The local maxima are at ( x , y ) = (x,y)= .
The local minima are at ( x , y ) = (x,y)= .
The saddle points are at ( x , y ) =

Given the function f (x, y) = ax^2
2 + 2xy + ay.y
2-ax-ay. Take
for a an integer value that is either greater than 1 or less
than -1, and
then determine the critical point of this function. Then
indicate whether it is
is a local maximum, a local minimum or a saddle point.
Given the function f (x, y) = ax^2 +2 + 2xy + ay^2-2-ax-ay.
Take
for a an integer value that is either greater than 1...

Problem 1.
(1 point)
Find the critical point of the function
f(x,y)=−(6x+y2+ln(|x+y|))f(x,y)=−(6x+y2+ln(|x+y|)).
c=?
Use the Second Derivative Test to determine whether it is
A. a local minimum
B. a local maximum
C. test fails
D. a saddle point

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