Question

4. When the second derivative test (SDT) is used for a critical point the result can be local maximum, local minimum, saddle point or test inconclusive. Consider the function f(x, y) = y 2 − 2y cos(x) with domain restricted to {(x, y) | −1 ≤ x ≤ 4 and − 2 ≤ y ≤ 4} and the three points P = ( π 2 , 0) Q = (0, 1) and R = (π, −1).

(a) Find all the critical points of f in the domain. The answer is ”P, Q, R and no others.” But you need to do the work to show that without knowing P, Q, R in advance.

(b) What does the SDT tell us about P ?

(c) What does the SDT tell us about Q ?

(d) What does the SDT tell us about R ? Make sure I see all the work needed to support your answers.

Answer #1

4. When the second derivative test (SDT) is used for a critical
point the result can be local maximum, local minimum, saddle point
or test inconclusive. Consider the function f(x, y) = y^2 − 2y
cos(x) with domain restricted to {(x, y) | −1 ≤ x ≤ 4 and − 2 ≤ y ≤
4} and the three points P = ( π/2 , 0) Q = (0, 1) and R = (π,
−1).
(a) Find all the critical points...

Find the critical point of the function f(x,y)=x2+y2+xy+12x
c=________
Use the Second Derivative Test to determine whether the point
is
A. a local maximum
B. a local minimum
C. a saddle point
D. test fails

If f(x,y)=(5∗x3+4∗y3+4∗x∗y+1) find the critical point for
f(x,y)
x=____
y=____
Is this critical point a local maximum, local minimum, or saddle
point?

Please use the second derivative test to determine whether each
point is local min/max or saddle of the function
f(x,y)=x^3-xy+y^3

Given f(x,y) = x2−3y2−8x+9y+3xy for each and any
point that is critical, use the second-partial-derivative test to
determine whether the point is a relative maximum, relative
minimum, or a saddle point.

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

Find the local maximum and minimum values and saddle point(s) of
the function. If you have three-dimensional graphing software,
graph the function with a domain and viewpoint that reveal all the
important aspects of the function. (Enter your answers as a
comma-separated list. If an answer does not exist, enter DNE.) f(x,
y) = 8 sin x sin y, −π < x < π, −π < y < π

Find the local maximum and minimum values and saddle point(s) of
the function. If you have three-dimensional graphing software,
graph the function with a domain and viewpoint that reveal all the
important aspects of the function. (Enter your answers as a
comma-separated list. If an answer does not exist, enter DNE.)
f(x, y) = 7 sin x sin y, −π < x
< π, −π < y < π

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical
point of the function f(x, y) = xy + y − x, and Q3 = 1 if f has a
local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at (Q1,
Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4
otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then T =...

Find the local maximum and minimum values and saddle point(s) of
the function. If you have three dimensional graphing software,
graph the function with a domain and viewpoint that reveal all the
important aspects of the function. (Enter NONE in any unused answer
blanks.) f(x, y) = 9y cos(x), 0 ≤ x ≤ π

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