Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the Junction at the critical points. f(x,y)=x^4y^2

Answers

Answer 1

Answer:

In a function of two variables, the critical points are these in which both derivatives are 0, and if:

[tex]\frac{d^2f}{dx^2}*\frac{d^2f}{dy^2} - [ \frac{d^2f}{dxdy} ]^2 < 0[/tex]

the point is a saddle point.

If the above equation is larger than zero, the critical points can be either a local maximum or a local minimum.

If it is equal to zero, no conclusions can be made.

First, let's find all these derivatives.

df/dx = 4*(x^3)*(y^2)

This is zero when x or y are equal to zero.

df/dy = 2*(x^4)*(y)

This is equal to zero when either x or y are equal to zero.

So in both cases, the points of the form:

(x, 0) and (0, y) are zeros of the first derivative, so these are the critical points.

Now let's look at the second derivatives:

[tex]\frac{d^2f}{dx} = 4*3*x^2*y^2 = 12*x^2*y^2[/tex]

[tex]\frac{d^2f}{dy^2} = 2*x^4[/tex]

[tex][\frac{d^2f}{dxdy} ]^2 = (4*x^3*2*y)^2 = 64*x^6*y^2[/tex]

Then, the equation to see if the points are saddle points or not are:

[tex](12*x^2*y^2)*(2*x^4) - 64*x^6*y^2 < 0[/tex]

In any of the previous found points (the ones of the form (x, 0) and (0, y)) the above equation will be equal to zero, then none of these points is a saddle point. More than that, in the particular case where the above equation is equal to zero, the critical points can be either: A relative minimum, a relative maximum, or a saddle point (so we need more analysis to find it out)

But in this case is ratter simple, because both powers in f(x, y) are even, we can conclude that the function is even and symmetric about x= 0 and y = 0, and the function is also positive, so all the points:

(0, y) and (x, 0) are the minimum values that f(x, y) can take.

Then the critical points are relative minimums.


Related Questions

consider the following exam scores from a class of 8 students


55,65,77,79,84,89.91,99


What is the rank of the student that scored a 77 on the exam?

Answers

Answer:

rank 6

Step-by-step explanation:

Some1 hurry need help

Answers

Answer:

Y=x/3 is the answer

Step-by-step explanation:

Hope this helps!!

WILL



GIVE BRAINLIST



ANSWER


MATH

Answers

Answer:

Mean = 4.7

Mode = 5

Median = 5

Step-by-step explanation:

Mean = Σfx / Σf

Σfx = (3*4) + (4*5) + (5*7) + (6*2) + (7*2) = 93

Σf = (4+5+7+2+2) = 20

Mean = 93 / 20

= 4.65

= 4.7

Mode = highest occurring = vlaue with highest frequency = 5

Median :

20 / 2 = 10

Value corresponding to the 10th term = 5

A random sample of 300 balls hit into play in college baseball games is taken. For each ball hit into play, it is recorded whether the ball was hit to the left, center, or right of the field, and whether the ball resulted in a hit or an out. Which of the following is the appropriate null hypothesis to test if there is a relationship between the direction of the ball and if it resulted in a hit or an out?
(A) H_0H0​: The proportion of balls hit to the left, right, and center is the same.
(B) H_0H0​: The proportion of balls that result in hits and outs is the same.
(C) H_0H0​: The distribution of the direction of the ball is the same for the population of hits and the population of outs.
(D) H_0H0​: The direction of the ball and whether it results in a hit or an out are independent.
(E) H_0H0​: There is an association between the direction of the ball and whether it results in a hit or an out.

Answers

Answer:

(D) H_0H0​: The direction of the ball and whether it results in a hit or an out are independent.

Step-by-step explanation:

Given that

There is 300 balls random sample is taken

We need to find out the appropriate null hypothesis that lies between the ball direction and lead to a hit or an out

So

Based on the given options, the option d is correct as the ball direction and its results either be hit or on out would represent that they are independent

Therefore the option d is correct

please help my teacher wont help

Answers

Answer: Choice B

They traveled 250 miles in 4 hours. Their speed was 62.5 mph.

=============================================================

Explanation:

Start at point X. Draw a horizontal line straight across until you reach the y axis. You should arrive at y = 250. This tells us that they traveled 250 miles.

Go back to point X. Draw a vertical line until you reach the x axis. You should reach x = 4. This indicates 4 hours.

Pairing up the two values together means they traveled 250 miles in 4 hours. Divide the distance over time to get the speed

speed = distance/time = 250/4 = 62.5 miles per hour

--------

Side note: unfortunately your teacher chose the label uppercase X for the point, which may be a bit confusing since it could clash with the lowercase x along the horizontal axis.

Another note: the speed represents the slope of the line

In a lab experiment, 300 bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every 18 hours. How many bacteria would there be after 3 hours, to the nearest whole number?

Answers

Answer:

337

Step-by-step explanation:

300(2)^(3/18)

= 300(2)^1/6

= 337.

Find the value of x.

Answers

Straight line: supplementary
Supplementary is sum of 180 degrees
x + 96 + 5x = 180 degrees
6x + 96 = 180
6x = 84, x = 14

Solution: x = 14

Answer:

x = 14

Step-by-step explanation:

Hope that will help you

The radius of a cylinder is 5 cm and the height is 7 cm.
Find the Lateral Area for the cylinder.
8011
A7011
12011
17511

Answers

Answer:

70 π

Step-by-step explanation:

Lateral Area of a Cylinder is A = 2πrh

Substitue the values in the above equation to get:

A = 2(π) * 5 * 7

A = 70 π

solve for x 5x 2^3x = 21

Answers

if u meant to put, 5x multipled by 2^3x=21 the answer is x= 0.798

Answer:

0.69

Step-by-step explanation:

someone please help me

Answers

Answer:

the answer is 35 mi

Step-by-step explanation:

28 squared = 784

21 squared = 441

784+441 = 1225

Square root of 1225 is 35

Carson = C

Hanford = H

Ridgewood = R

CR²=CH²+HR² <=>

<=> CR²=28²+21²=1225 <=> CR=√1225=35

Jennifer has 84.5 yards of fabric to make curtains for her new house. She made
identical curtains and has 19.7 yards of fabric left over.
How many yards of fabric does Jennifer use for each curtain?

Answers

Answer:

64.8

Step-by-step explanation:

84.5 - 19.7 = 64.8

Find the coordinates of the point P that divides the directed line segment from A to B in the given ratio.
A (8,6), B(-2,-9); 3 to 2
The coordinates of point P are



Answers

Answer:

[tex]P = (2,-3)[/tex]

Step-by-step explanation:

Given

[tex]m : n = 3 : 2[/tex]

[tex]A = (8,6)[/tex]

[tex]B = (-2,-9)[/tex]

Required

The coordinate of {

This is calculated as:

[tex]P = (\frac{mx_2 + nx_1}{m + n},\frac{my_2 + ny_1}{m + n})[/tex]

So, we have:

[tex]P = (\frac{3*-2 + 2*8}{3 + 2},\frac{3*-9 + 2*6}{3 + 2})[/tex]

[tex]P = (\frac{10}{5},\frac{-15}{5})[/tex]

[tex]P = (2,-3)[/tex]

HELP ME AND GIVE ME THE ANSWER I DONT NEED U TO EXPLAIN ANYTHING JUST HURRY UP

Answers

Answer:

x = 9.1

Step-by-step explanation:

What is 34×67 what is it

Answers

Answer:

2278

Step-by-step explanation:

What is the z score associated with sleeping 503 minutes per night

Answers

Answer:

you need to mutiply zero meana503 neary by 500

In a psychology class, 60 students have a mean score of 96.9 on a test. Then 19 more students take the test and their mean score is 71.3.

What is the mean score of all of these students together? Round to one decimal place.
Help please, and thank you! C:

Answers

Answer:83.3

Step-by-step explanation:

plz i need help asappppppppp

Answers

i think the answer is -5???

Answer:

-10

Step-by-step explanation:

a bag contains 48 balls
the ratio of red to blue balls is 3:5 find how many red and blue balls there are.
please help ​

Answers

Answer:

25 Balls

Step-by-step explanation:

Does the quadratic function
y = 3x2 + 7x - 8
have one, two or no real zeros?
Utilize the quadratic formula to
determine the answer.
[?] real zeros

Answers

Answer:

Two real roots

Step-by-step explanation:

The portion of the quadratic formula that determines the number of roots is

b² - 4ac

If it's positve there are two real roots

If it's zero there is one real root

If it's negative there are no real roots

------------------------------

b² - 4ac = 7² - 4(3)(-8)

b² - 4ac =  49 + 96

b² - 4ac =  145

Two real roots since 145 is positive

The quadratic equation has two real zeroes

What is quadratic equations?

The polynomial equations of degree two in one variable of type f(x) = ax2 + bx + c = 0 and with a, b, c, and R R and a 0 are known as quadratic equations. It is a quadratic equation in its general form, where "a" stands for the leading coefficient and "c" for the absolute term of f. (x). The roots of the quadratic equation are the values of x that fulfill the equation (, ).

It is a given that the quadratic equation has two roots. Roots might have either a real or imaginary nature.

What is quadratic formlua?

D=b²-4ac

x=(-b±√(b²-4ac))/2a

y = 3x² + 7x - 8

On comparing with ax²+bx+c, we get

a=3, b=7 and c=-8

Using Quadratic formula

D=b²-4ac

   =7²+4x3x8

   =49+96

     =145>0

Hence it has real zeroes

Learn more about Quadratic equation,

https://brainly.com/question/776122

#SPJ2

Find the volume of the square pyramid shown (please help)

Answers

Answer:

Given:

     b = 8 cm

     h = 3 cm

Find : Volume

Formula: V = [tex]a^{2}[/tex]h/3

Solution: V = (8 cm)2 ( 3 cm)/ 3

               V = (64 cm2)(3 cm)/3

               V = 192 cm3/3

               V = 64 cubic cm

Rocco has a garden and an oak tree. The oak tree is 11 times as old as the garden. The sum of their ages is 108 years. How old is each the garden and tree?

Answers

Answer:

GARDEN = 9 years

TREE = 99 years

Step-by-step explanation:

the following equation can be derived from the question

age of oak tree = 11x

age of garden = x

11x + x = 108

12x = 108

x = 9

oak tree = 11 x 9 = 99

In the figure below, points A, B, and C are ______ points because they lie on the same line.

Answers

Answer:

A, B, C are Collinear points

Step-by-step explanation:

Find tano, where is the angle shown.
Give an exact value, not a decimal approximation.
17
ө
8

Answers

Answer:

Step-by-step explanation:

cos θ=8/17

sec θ=17/8

sec² θ-tan²θ=1

tan θ=√(sec²θ-1)=√((17/8)²-1)=√((289-64)/64)=√(225/64)=15/8

A used bookstore is having a sale. All paperback books
are $0.75 and all hardcover books are $3. Ginny has $10.
She wants to buy one hardcover book and as many
paperback books as she can. Which inequality represents
this situation, where p represents the number of paperback
books Ginny can buy?

Answers

Answer:

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(Gotta get with my friends)

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So tell me what you want, what you really, really want

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If you wanna be my lover, you gotta get with my friends

(Gotta get with my friends)

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So, here's a story from A to Z

You wanna get with me, you gotta listen carefully

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If you wanna be my lover

Step-by-step explanation:

Need help asap please

Answers

Answer:

58

Step-by-step explanation:

50 POINTS !!


PLEASE HELP

EXPLAIN HOW YOU GOT IT !!!!

ILL GIVE BRAINLIEST TO THE RIGHT ANSWERS.

Answers

Answer:

Step-by-step explanation:

Hope this helps u!!

Which Number has the highest value?

-25

16

-32

34

Answers

Answer:

34

Step-by-step explanation:

Answer:

34

Step-by-step explanation:

its the largest, positive number

Find the total cost of $16 IMax movie with you have a 15% student
discount. *

Answers

Answer:

$13.60

Step-by-step explanation:

16.00 x .85 = 13.60$

-- Gage Millar, Algebra 2 Tutor

Answer:

$18.40 is the final cost

Explanation:

Multiply 15% and 16 and add the product with 16.


Someone said the answer is -5 but can you explain how?

Answers

Answer:

-5

Step-by-step explanation:

add 5 and 10 makes 14

minus 5 from 15

multiply 10 and 4 and add to 15 to get 40

subtract 45 from 40 and get -5

he started with $5 then gets $10 so he then had $15 but he bought a toy which was $5 so $15-$5= so after that he had $10. On his birthday he receives $10 from 3 of his friends so $30 in total plus the $10 he had from before and he had to pay $45 but he only had $40 therefore the answer is -5.

m and n are parallel lines cut by a transversal.
What is the value of x + y?

Answers

Answer:

180°

Step-by-step explanation:

It will be answer ...............

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