Multivariable optimization question. Find three positive real numbers whose sum is one and the sum of their squares is a minimum.

boitshupoO 2021-02-19 Answered
Multivariable optimization question. Find three positive real numbers whose sum is one and the sum of their squares is a minimum.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

2k1enyvp
Answered 2021-02-20 Author has 94 answers

Let the three positive real numbers be x,y and z.
then, x+y+z=1
let sum of squares be:
f(x,y,z)=x2+y2+z2
put z in above function
g(x,y)=x2+y2+(1(x+y))2
=x+y2+1+(x+y)221(x+y)
=x2+y2+1+22+y2+2xy2x2y
=2x2+2y2+2xy2x2y+1
Partially differenting the function g(x,y) and finding the critical points:
gx=4x+2y2=0
gy=4y+2x2=0
From the above two equtioan, we get:
x=13,y=13
Putting the value of x and y in x+y+z=1, we get, z=12
Using the second derivative test:
g×=4,
gyy=4,
gxy4x+2=4(13)+2=103
D=gxxgyygxy2
=44(103)2
=161009
=449>0
Since gxx>0andD>0,p(13,13,13) is a local minimal
Therefore, the sum of squarers will be minimom if x=y=z=13 Hence, x=13,y=13,z=13.

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-02-05
Use polar coordinates to find the limit. [Hint: Let x=rcosandy=rsin , and note that (x, y) (0, 0) implies r 0.] lim(x,y)(0,0)x2y2x2+y2
asked 2021-09-08
Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.
P(x)=12x2+2136x41000
To maximize the monthly rental profit , how many units should be rented out?
What is the maximum monthly profit realizable?
asked 2020-11-27
Comsider this, multivariable fucntion
f(x,y)=3xyx2+5y225
a)What is the value of f(-1,3)?
b)Find all x-values such that f(x,x)=0
asked 2021-11-22
Find the solution set for the system
x2+y2=25
25x2+y2=25
by graphing both of the system’s equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.
asked 2021-09-05

Use the following linear regression equation to answer the questions.
x1=1.3+3.0x28.3x3+1.6x4
(a) Which variable is the response variable?
(b) Which number is the constant term? List the coefficients with their corresponding explanatory variables.
(c) If x2=8,x3=2,andx4=6 , what is the predicted value for x1? (Use 1 decimal place)
(d) Explain how each coefficient can be thought of as a "slope" under certain conditions.

asked 2021-11-22
A commodity has a demand function modeled by p=1260.5x and a total cost function modeled by C=50x+39.75, where x is the number of units.
(a)Use the first marginal analysis criterion presented in class to find the production level that yields a maximum profit. Then use the second criterion to ensure that level yields a maximum and not a minimum. Finally, what unit price (in dollars) yields a maximum profit?
$ per unit
(b)When the profit is maximized, what is the average total cost (in dollars) per unit? (Round your answer to two decimal places.)
$ per unit
asked 2021-09-16
Find the complex zeros of the following polynomial function. Write f in factored form.
f(x)=x312x2+49x58
The complex zeros of f are =?