1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Andrew [12]
3 years ago
9

Local versus absolute extrema. If you recall from single-variable calculus (calculus I), if a function has only one critical poi

nt, 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
Mathematics
1 answer:
stich3 [128]3 years ago
7 0

Answer:

Step-by-step explanation:

Given that:

a)

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

\text{Now; to determine the critical point:}- f_x = 0 ;  \ \ \ \ \  f_y =0

\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)

b)

\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

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  \\ \\

\implies t_{0,1} = 6

Now; D = rt - s^2 \\ \\  = (6)(6) -(-3)^2

= 36 - 9 \\ \\ = 27 > 0 \\ \\  r>0

\text{Hence, the critical point} \ (0,1)  \ \text{appears to be the local minima}

c)

\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

\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)}

You might be interested in
Determine whether the following is a statistical question.
-BARSIC- [3]

Answer:

Yes, the given question is a statistical question.

Step-by-step explanation:

Given: statement is "What is the typical height of dog kennels at Keita's Kennels?"

To check: whether the given statement is a statistical question

Solution:

A statistical question is one for which you will generally get more than one answer.

For example "What's the age of the students in your school?" is a statistical question but "What's your age?" is not a statistical question.

The given statement "What is the typical height of dog kennels at Keita's Kennels?" has a single answer only, so the given question is statistical

7 0
3 years ago
PLEASE PLEASE PLEASE PLEASE PLEASE PLEASE Please help please
umka21 [38]
The correct answer is:  "(-1, -2)" .
________________________________________
Note:  By examining the point of intersection of the lines of the 2 graphs show in the "image attached", we can see that the point of intersection is at:  "(-1, 2");  that is:  "x = -1, y = 2" .
________________________________________
3 0
3 years ago
Read 2 more answers
5d-2/3c if d =4 and c =3 evaluate fast fast
Mademuasel [1]

Answer:

2

Step-by-step explanation:

Plug 4 in for d and 3 in for c

5d-2/3c

5(4)-2/3(3)

Multiply in the numerator and the denominator

20-2/9

Subtract in the numerator

18/9

2

Hope this helps! :)

4 0
3 years ago
House is 50.0 ft long and 26 ft wide and has 8.0-ft- m high ceilings. what is the volume of the interior of the house in cubic m
Tom [10]
560 cm to the power of 2 
3 0
3 years ago
Mrs. Elizondo runs around a 400-meter track at a constant speed of 250 meter per minute. How many minutes does it take Mrs. Eliz
anastassius [24]

Answer: See explanation

Step-by-step explanation:

From the information, we are told that Mrs. Elizondo runs around a 400-meter track. If she runs 4 laps, the total meters will be:

= 400 × 4

= 1600 meters

Since she uses a constant speed of 250 meter per minute. The number if minutes taken to complete the 4 laps will be gotten by dividing 1600 by 250. This will be:

= 1600 / 250

= 6.4 minutes

= 6 minutes + (0.4 × 60 seconds).

= 6 minutes 24 seconds

3 0
3 years ago
Other questions:
  • Plot 43 and 263 on the number line.
    13·2 answers
  • Kite QRST has a short diagonal of QS and a long diagonal of RT. The diagonals intersect at point P. Side QR= 5 m and diagonal QS
    13·2 answers
  • Find the value of x. If necessary, round to the nearest tenth.
    10·1 answer
  • What are the steps for using a compass and straightedge to construct the bisector of ZA?
    7·2 answers
  • The length of a rectangle is 72 cm and the diagonal measures 75 cm. Find the width and perimeter of the rectangle.
    8·1 answer
  • Jenna and her 3 friends each picked 3 apples off the apple tree in her yard. If an apple weighs 5 ounces, what is the total weig
    12·2 answers
  • PLEASE HELP ASAP.
    5·1 answer
  • Use proportional reasoning to find 46% of 50. Show how you set up the proportion.
    7·1 answer
  • Supervisor: "You have been averaging 55 sales per day. Per our new policy, everyone
    8·1 answer
  • Which two elements have similar characteristics<br><br> Plz answer right hry
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!