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
pantera1 [17]
1 year ago
8

The sum of two positive numbers is 16. What is the optimum value (maximum or minimum) for the sum of their squares?

Mathematics
1 answer:
Nikitich [7]1 year ago
8 0

x=8 and y=8 are the two positive integers whose sum is 16 and sum of their squares is minimum.

What is optimum value ?

The optimum value is a minimum or maximum value of the objective function over the feasible region of an optimization problem.

If a function is strictly increasing in a definite interval and increases up to a fixed value and after this, it starts decreasing, then that point is called maximum point of the function and value of function at that point is called maximum value.

If a function is strictly decreasing in a definite interval and decreases up to a fixed value and after this, it starts increasing, then that points is called minimum point of the function and the value of function at that point is called minimum value.

Conditions for finding maxima and minima

The conditions for maxima and minima for a function y=f(x) at a point x=a  are as follow:

1. Necessary condition

for maxima and minima, the necessary condition is

f'(x)=\frac{dy}{dx}

2.Suffiecient condition

for maxima and minima, the  necessary condition are

for maximum value

at x=a,\frac{d^2y}{dx^2} should be negative.

for minimum value

at x=a, \frac{d^2y}{dx^2} should be positive.

The sum of two positive number is 16.

We have to find the maximum and minimum value for the sum of their squares.

The sum of two positive number is 16.

let the number be x and y, such that x > 0 and y > 0

sum of the number is x+y=16

sum of squares of the number S=x^2+y^2

x+y=16\\y=16-x ----------1\\S=x^2+y^2\\S=x^2+(16-x)^2-----------2after substituting the value of y from equation 1

for finding the maximum and minimum of given function we can find it by differentiating the function with x<em> </em>equal it to 0

Differentiate the equation 2

\frac{dS}{dx} =\frac{d}{dx}[x^2+(16-x)^2]\\\frac{dS}{dx}=\frac{d}{dx}(x^2)+\frac{d}{dx}(16-x^2)\\ \frac{dS}{dx}=2x+2(16-x)(-1)---------3

Now equating the first derivative equal to zero

so, \frac{dS}{dx}=0

2x+2(16-x)(-1)=0\\2x-2(16-x)=0\\2x-32+2x=0\\4x-32=0\\4x=32\\x=\frac{32}{4}=8

As x > 0, x=8

Now, for checking if the value of S is minimum or maximum at x=8, we will perform the second derivative of S with respect to x

\frac{d^2S}{dx^2}=\frac{d}{dx}[2x+2(16-x)(-1)]\\\frac{d^2S}{dx^2}=\frac{d}{dx}[2x-2(16-x)]\\\frac{d^2S}{dx^2}=\frac{d}{dx}(2x)-2\frac{d}{dx}(16-x)\\\frac{d^2S}{dx^2}=2-2(0-1)\\\frac{d^2S}{dx^2}=2-0+2=4\\\frac{d^2S}{dx^2}=4

According to the sufficient condition if the second derivative is positive then the value is minimum

hence for x=8 will be the minimum point of the function S.

Therefore the function S sum of squares of the two number is minimum at x=8

from equation 1

y=16-x\\y=16-8\\y=8

Therefore , x=8 and y=8 are the two positive numbers whose sum is 16 and the sum of their squares is minimum.

Learn more about the optimum value (maximum or minimum) here

brainly.com/question/28284783

#SPJ4

You might be interested in
Nancy had 333 dollars to spend on 9 books. After buying them she had 18 dollars. How much did each book cost ?
MissTica

Answer:

35 dollars

Step-by-step explanation:333-18=315 dollars

315/9=35 dollars for each book

7 0
3 years ago
Determine whether The given segments have the same length. Justify your answer
Semenov [28]

We know the distance formula is

\sqrt{ (x_{2}-x_{1})^2+ (y_{2}-y_{1})^2 }

9)

Here A( -4,2) and B(1,4)

So length of AB

= \sqrt{(1-(-4))^2+(4-2)^2} =\sqrt{5^2+2^2} =\sqrt{29}

Also C(2,1)

Length of BC

= \sqrt{(2-1)^2+(-1-4)^2} =\sqrt{1^2+(-5)^2} =\sqrt{26}

So we can see that length of AB is not equal to length of BC

11.

Now AB = \sqrt{29}

Also C(2,-1) & D(4,4)

Length of CD

= \sqrt{(4-2)^2+(4-(-1)) ^2} =\sqrt{2^2+5^2} =\sqrt{29}

Yes AB = CD

5 0
3 years ago
A mountain climber ascended 3/4 of a kilometer in 1/12 of an hour. What was the mountain climber’s ascent rate in kilometers per
Yanka [14]
<span>A mountain climber ascended 3/4 of a kilometer in 1/12 of an hour. What was the mountain climber’s ascent rate in kilometers per hour? 

Correct Answer: C. 9 Kilometers per hour </span>
3 0
3 years ago
Read 2 more answers
Thank you so much to anyone who answers this question!
Anna71 [15]

C. The 40th customer

4 0
2 years ago
Keith bought 4 shirts for $36 at this rate what would his cost be for 7 shirts
Norma-Jean [14]

Answer:

$63

Step-by-step explanation:

we know that

Keith bought 4 shirts for $36

so

using proportion

Find out would be the cost for 7 shirts

Let

x ----> the cost for 7 shirts

\frac{4}{36}\ \frac{shirts}{\$}=\frac{7}{x}\ \frac{shirts}{\$}\\\\x=36(7)/4\\\\x=\$63

3 0
3 years ago
Other questions:
  • Simplify 6√1,000m^3n^12
    11·2 answers
  • The two-column proof below describes the statement and reason for proving that corresponding angles are congruent.
    5·2 answers
  • What is true regarding secants and chords
    10·2 answers
  • Suppose a life insurance company sells a ​$250 comma 000 ​one-year term life insurance policy to a 23​-year-old female for ​$190
    10·1 answer
  • Evaluate 3 - 24 + 8 + 42.<br> A.0<br> B.2<br> C.16<br> D.24
    13·2 answers
  • I’m not sure how to solve this. help?
    13·1 answer
  • Expresa algebraicamente como varia el costo en cada uno de los hoteles al aumentar la estancia Confecciona, para cada uno de los
    15·1 answer
  • Find the area please :)))
    14·1 answer
  • What is 8993838383 x 4?
    15·1 answer
  • I’ll give 20 points for the ans and equation.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!