Local versus absolute extrema. If you recall from single-variable calculus (calculus I), if a function has only one critical point, and that critical point is a local maximum (or say local minimum), then that critical point is the global/absolute maximum (or say global/absolute minnimum). This fails spectacularly in higher dimensions (and thereís a famous example of a mistake in a mathematical physics paper because this fact was not properly appreciated.) You will compute a simple example in this problem. Let f(x; y) = e 3x + y 3 3yex . (a) Find all critical points for this function; in so doing you will see there is only one. (b) Verify this critical point is a local minimum. (c) Show this is not the absolute minimum by Önding values of f(x; y) that are lower than the value at this critical point. We suggest looking at values f(0; y) for suitably chosen y
Answer:
Step-by-step explanation:
Given that:
a)
[tex]f(x,y) = e^{3x} + y^3 - 3ye^x \\ \\ \implies \dfrac{\partial f}{\partial x} = 0 = 3e^{3x} -3y e^x = 0 \\ \\ e^{2x}= y \\ \\ \\ \implies \dfrac{\partial f}{\partial y } = 0 = 3y^2 -3e^x = 0 \\ \\ y^2 = e^x[/tex]
[tex]\text{Now; to determine the critical point:}[/tex]- [tex]f_x = 0 ; \ \ \ \ \ f_y =0[/tex]
[tex]\implies e^{2x} = y^4 = y \\ \\ \implies y = 0 \& y =1 \\ \\ since y \ne 0 , \ \ y = 1, \ \ x= 0\\\text{Hence, the only possible critical point= }(0,1)[/tex]
b)
[tex]\delta = f_xx, s = f_{xy}, t = f_{yy} \\ \\ . \ \ \ \ \ \ \ \ D = rt-s^2 \\ \\ i) Suppose D >0 ,\ \ \ r> 0 \ \text{then f is minima} \\ \\ ii) Suppose \ D >0 ,\ \ \ r< 0 \ \text{then f is mixima} \\ \\ iii) \text{Suppose D} < 0 \text{, then f is a saddle point} \\ \\ iv) Suppose \ D = 0 \ \ No \ conclusion[/tex]
[tex]Thus \ at (0,1) \\ \\ \delta = f_{xx} = ge^{3x}\implies \delta (0,1) = 6 \\ \\ S = f_{xy} = -3e^x \\ \\ \implies S_{(0,1)} = -3 \\ \\ t = f_{yy} = 6y \\ \\[/tex]
[tex]\implies t_{0,1} = 6[/tex]
[tex]Now; D = rt - s^2 \\ \\ = (6)(6) -(-3)^2[/tex]
[tex]= 36 - 9 \\ \\ = 27 > 0 \\ \\ r>0[/tex]
[tex]\text{Hence, the critical point} \ (0,1) \ \text{appears to be the local minima}[/tex]
c)
[tex]\text{Suppose we chose x = 0 and y = -3.4} \\ \\ \text{Then, we have:} \\ \\ f(0,-3.4) = 1+ (-3.4)^3 + 3(3.4) \\ \\ = -28.104 < -1[/tex]
[tex]\text{However, if f (0,1) = 1 +1 -3 = -1 \\ \\ f(0,-3.4) = -28.104} < -1} \\ \\ \text{This explains that} -1 \text{is not an absolute minimum value of f(x,y)}[/tex]
is 2/3 equal to 2/4 ? if not is it greater
Answer:
Equivalent fraction of a given fraction is got by multiplying or dividing its numerator and denominator by the same whole number. For example, if we multiply the numerator and denominator of 2/3 by 4 we get. 2/3 = 2×4 / 3×4 = 8/12 which is an equivalent fraction of 2/3.
Step-by-step explanation:
Owen runs 37 miles every 2 weeks. Enter the number of miles Owen runs in 12 weeks
John buys an item that costs $50.00 is marked 20% off. Sales tax for the item is 8%. What is the final price that John pays, including tax?
Answer:
$43.20
Step-by-step explanation:
20% of $50.00 is $10.00. $50 - $10 is $40. 8% of $40 is $3.20. Since this is tax, you must add it. $40+$3.20 = $43.20
A summer movie club charges a $30
membership fee and then $5 per movie.
Which equation can be used to find y, the
total cost of the movie club, if x represents
the number of movies watched.
Is B right?
one of the fastest times for 1,500-meter race is 3 minutes and 34 seconds. How many seconds is this time?
Answer:
214 seconds
Step-by-step explanation:
3 times 60 is 180. 180 plus 34 is 214.
In Ms. Chance's first period class, there are 9 students with hazel eyes, 10 with brown eyes, 7 with blue eyes, and 2 students with green eyes. Ms. Chance picks a student at random. Which color eyes is the student most likely to have?
What are the coordinates of point A
Answer:
The coordinates of point A are (-4, 3)
hope it helps
Answer:
The coordinates of point A is (-4,3)
Step-by-step explanation:
(x,y)
X= -4
Y= 3
B
7
25
с
24
А
Find sin(a) in the triangle.
Answer: 7/25
Step-by-step explanation:
got it correct on khan academy
Is the following relation a function? *
{(-3, 2), (1, 8), (-1, 5), (3, 11)}
Yes
No
Why or why not?
Answer:
No.
Step-by-step explanation:
Well at first glance it might seem so but there are two points particularly that can tell you that these points cannot be within a function.
The points (3,2) and (3,-2) will yield an undefined slope (a straight vertical line). There is no possibility that the other points can be in this line (as their y - values are different) and there is no possibilty that this is a function at all according to the vertical line test (the test is that if you draw a vertical line that there shouldn't be more than one point on it).
Select the option that best describes the relationship
between the variables on the scatter plot.
35
positive, linear association
30
25
no association
20
15
positive, non-linear association
10
5
negative, linear association
10
12
Activate Windows
Answer:
4th one its negative,linear association
Step-by-step explanation:
The option that best describes the relationship between the variables on the scatter plot is negative, linear association.
What is a negative linear association?Variables have a negative association when the line of best fit is downward sloping. Variables have a linear association when the line of best fit is a straight line. Variables have a positive association when the line of best fit is upward sloping.
To learn more about linear functions, please check: https://brainly.com/question/26434260
David borrows $15,000 for a new car. He is
charged 6% simple interest per year. How much
interest is charged for 5 years?
2^3=4^p find the value of p
Answer:
Exact form: p= 3/2
Decimal form: p= 1.5
Mixed Number: p=1 [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
The following data show the average retirement ages for a random sample of workers in the United States and a random sample of workers in Japan. Perform a hypothesis test using α = 0.05 to determine if the average retirement age in Japan is different from the United States. Calculate the test-statistics (round to 3 decimals - report the absolute value).
Answer:
[tex]t \approx 2.639[/tex]
Step-by-step explanation:
Given
[tex]\begin{array}{ccc}{} & {USA\ 1} & {Japan\ 2} & {\bar x} & {64.6} & {67.5} &{n} & {30} & {30} & {\sigma} & {4.0} & {4.5} \ \end{array}[/tex]
See attachment for data
Required
Determine the test statistic
The test statistic is calculated using:
[tex]t = \frac{\bar x_1 - \bar x_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}[/tex]
So, we have:
[tex]t = \frac{64.6 - 67.5}{\sqrt{\frac{4.0^2}{30} + \frac{4.5^2}{30}}}[/tex]
[tex]t = \frac{64.6 - 67.5}{\sqrt{\frac{16.00}{30} + \frac{20.25}{30}}}[/tex]
[tex]t = \frac{64.6 - 67.5}{\sqrt{\frac{16.00+20.25}{30}}}[/tex]
[tex]t = \frac{64.6 - 67.5}{\sqrt{\frac{36.25}{30}}}[/tex]
[tex]t = \frac{-2.9}{\sqrt{1.2083}}[/tex]
[tex]t = \frac{-2.9}{1.099}[/tex]
[tex]t \approx -2.639[/tex]
The absolute value is:
[tex]t \approx 2.639[/tex]
How to show work on this one topic is angles formed by 2 chords and secants
Answer:
Step-by-step explanation:
Need the answer to x and y !
Answer:
x=5, y=3
Step-by-step explanation:
7x-11=24: 7x=35: x=5
3(5)+9=8x: 24=8x: x=3
On his last 5 math tests, Gilbert earned: 94, 72, 83, 94, 90. What was his average score? ( Report your answer to the tenths place).
Answer:
Average of Gilbert in last 5 test = 86.6
Step-by-step explanation:
Given:
Score earned by Gilbert in last 5 test = 94, 72, 83, 94, 90
Find:
Average of Gilbert in last 5 test
Computation:
Average value = Sum of all events / Total number of events
Average of Gilbert in last 5 test = Sum of all last results / Number of results
Average of Gilbert in last 5 test = [94 + 72 + 83 + 94 + 90] / 5
Average of Gilbert in last 5 test = [433] / 5
Average of Gilbert in last 5 test = 86.6
A research company desires to know the mean consumption of meat per week among males over age 43. A sample of 1384 males over age 43 was drawn and the mean meat consumption was 3 pounds. Assume that the population standard deviation is known to be 1.3 pounds. Construct the 99% confidence interval for the mean consumption of meat among males over age 43. Round your answers to one decimal place.
Answer:
The 99% confidence interval for the mean consumption of meat among males over age 43 is between 2.9 pounds and 3.1 pounds.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1 - 0.99}{2} = 0.005[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1 - \alpha[/tex].
That is z with a pvalue of [tex]1 - 0.005 = 0.995[/tex], so Z = 2.575.
Now, find the margin of error M as such
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 2.575\frac{1.3}{\sqrt{1384}} = 0.1[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 3 - 0.1 = 2.9 pounds
The upper end of the interval is the sample mean added to M. So it is 3 + 0.01 = 3.1 pounds
The 99% confidence interval for the mean consumption of meat among males over age 43 is between 2.9 pounds and 3.1 pounds.
The radius of a circle is 16 cm. Find its area in terms of \piπ
Answer256pi
Step-by-step explanation:
Area=pi*r^2
=pi*16^2
256pi
help me please!! i’m confused!!
Evaluate the algebraic expression for the given value(s) of the variable(s).
2x 2 - 7xy + 3y 2 ; x = -3, y = 4
Answer:
-18
Step-by-step explanation:
2x²-7xy+3y²
2(3²)-7(3)(4)+3(4²)
2(9)-84+3(16)
18-84+48
18+48-84
-18
You want to have $150,000 in your retirement account when you retire in 30 years. Your retirement account earns 7%
interest. How much do you need to deposit each month to meet your retirement goal?
Round your answer to the nearest cent.
Do NOT include the dollar sign.
Answer: 123.05
Use annuity formula
Answer:
Step-by-step explanation:
Since we're talking about making a deposit of a certain amount every month rather than just one big deposit, we are talking about an annuity. The formula for the value of an annuity is A(t)=d[(1+rn)nt−1](rn) where A(t) is the value of the annuity, d is the amount of each deposit, n is the number of deposits per year, t is the number of years, and r is the rate of interest. In this case we know we want the value of the annuity to be A(t)=$150,000, we want to make deposits every month, or 12 times a year, so n=12, we want to reach our desired value in 30 years, so t=30, and our account earns 7% interest, so r=0.07. We can plug in all of these values and solve for d to find the amount we need to deposit each month:
A(t)150,000150,000150,000d=d[(1+rn)nt−1](rn)=d[(1+0.0712)12⋅30−1]0.0712≈d[(1.00583)360−1]0.00583≈1,219.97d≈122.95
The amount you need to deposit each month is approximately $122.95.
if d is 55 degrees then what is F and E degrees. Explain how you know
Answer:
F=55
E=70
Step-by-step explanation:
The total degree of a triangle is 180.
So we have 180-55*2
E=180-110
E=70
The hanger image below represents a balanced equation. Write an equation to represent the image. PLEASE HELP ILL MARK U AS BRAINLIST OR WHATEVER o(*°▽°*)o
Answer:
i think its 3 in all the boxes but trying is better than not ig wait no nvm
what is the remainder when the polynomial f(x) = x³ - x² + 3x - 2, is divided by 2x - 1
Answer:
Equate the divisor to 0
2x-1=0
2×=1
×=1/2
Putting onto the polynomial
f(1/2) = (1/2)³-1/2)²+3(1/2)-2
=-5/8
5 divided by 1/4
go straight to the answer
Answer:
20
Step-by-step explanation:
(Sorry for my brothers dirty ipad-) but can someone answer this for him? points: 52
Answer:
12 students
Step-by-step explanation:
CLEAN THAT IPAD FOR HIM
Answer:
12 students went to the beach fewer than 2 times.
Step-by-step explanation:
Just count the total number of dots that come before 2.
Brainliest goes to whoever explains the answer best pls help
Answer: (-2,9)
Step-by-step explanation: I did this not too long ago!
Solve equation [2] for the variable y
[2] y = 5x + 19
// Plug this in for variable y in equation [1]
[1] (5x+19) + 3x = 3
[1] 8x = -16
Solve equation [1] for the variable x
[1] 8x = - 16
[1] x = - 2
By now we know this much :
y = 5x+19
x = -2
Use the x value to solve for y
y = 5(-2)+19 = 9
Answer (-2,9)
Vincent began his weekly chores on Saturday morning at 11:20 he worked for 1 hour and 15 minutes with a 10 minute break at what time did Vincent finish his chores
Answer:
12:45
Step-by-step explanation:
Step One: An hour after 11:20 is 12:20, plus 15 minutes is 12: 35.
Step Two: Just add 10 minutes, because it doesn't matter when you add the ten minutes.
WILL GIVE BRAINLIEST IF SOLVED IN 10 MINS : Solve each equation with the quadratic :
Answer:
5. x = -1 or x = -3
6. b = -2 or b = 6
7. n = 5 or n = 4
Step-by-step explanation:
5. x² + 4x + 3 = 0
Factorize
x² + 3x + x + 3 = 0
x(x + 3) +1(x + 3) = 0
(x + 1)(x + 3) = 0
Set each of the factors equal to zero and solve for x
x + 1 = 0 or x + 3 = 0
x = -1 or x = -3
6. b² - 4b - 14 = -2
Put all terms on one side of the equation leaving zero on one side
b² - 4b - 14 + 2 = 0
b² - 4b - 12 = 0
Factorize
b² - 6b + 2b - 12 = 0
b(b - 6) +2(b - 6) = 0
(b + 2)(b - 6) = 0
Set each of the factors equal to zero and solve for b
b + 2 = 0 or b - 6 = 0
b = -2 or b = 6
7. n² = 9n - 20
Put all terms on one side of the equation leaving zero on one side
n² - 9n + 20 = 0
n² - 5n - 4n + 20 = 0
n(n - 5) -4(n - 5) = 0
(n - 5)(n - 4) = 0
Set each of the factors equal to zero and solve for b
n - 5 = 0 or n - 4 = 0
n = 5 or n = 4