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soldier1979 [14.2K]
3 years ago
11

Use a whole number, a fraction, one plus sign, and four 2's to make 23.

Mathematics
1 answer:
maksim [4K]3 years ago
7 0
(2+1/2-1/2)x2x2x2+7=23
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. Use Lagrange multipliers to find the maximum and minimum values of the function, f, subject to the given constraint, g. (Place
zzz [600]

Answer:

Minimum value of f(x, y, z) = (1/3)

Step-by-step explanation:

f(x, y, z) = x⁴ + y⁴ + z⁴

We're to maximize and minimize this function subject to the constraint that

g(x, y, z) = x² + y² + z² = 1

The constraint can be rewritten as

x² + y² + z² - 1 = 0

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

where λ = Lagrange factor, which can be a function of x, y and z

L(x,y,z) = x⁴ + y⁴ + z⁴ - λ(x² + y² + z² - 1)

We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points, each of the partial derivatives is equal to 0.

(∂L/∂x) = 4x³ - λx = 0

λ = 4x² (eqn 1)

(∂L/∂y) = 4y³ - λy = 0

λ = 4y² (eqn 2)

(∂L/∂z) = 4z³ - λz = 0

λ = 4z² (eqn 3)

(∂L/∂λ) = x² + y² + z² - 1 = 0 (eqn 4)

We can then equate the values of λ from the first 3 partial derivatives and solve for the values of x, y and z

4x² = 4y²

4x² - 4y² = 0

(2x - 2y)(2x + 2y) = 0

x = y or x = -y

Also,

4x² = 4z²

4x² - 4z² = 0

(2x - 2z) (2x + 2z) = 0

x = z or x = -z

when x = y, x = z

when x = -y, x = -z

Hence, at the point where the box has maximum and minimal area,

x = y = z

And

x = -y = -z

Putting these into the constraint equation or the solution of the fourth partial derivative,

x² + y² + z² = 1

x = y = z

x² + x² + x² = 1

3x² = 1

x = √(1/3)

x = y = z = √(1/3)

when x = -y = -z

x² + y² + z² = 1

x² + x² + x² = 1

3x² = 1

x = √(1/3)

y = z = -√(1/3)

Inserting these into the function f(x,y,z)

f(x, y, z) = x⁴ + y⁴ + z⁴

We know that the two types of answers for x, y and z both resulting the same quantity

√(1/3)

f(x, y, z) = x⁴ + y⁴ + z⁴

f(x, y, z) = (√(1/3)⁴ + (√(1/3)⁴ + (√(1/3)⁴

f(x, y, z) = 3 × (1/9) = (1/3).

We know this point is a minimum point because when the values of x, y and z at turning points are inserted into the second derivatives, all the answers are positive! Indicating that this points obtained are

S = (1/3)

Hope this Helps!!!

6 0
2 years ago
Will give Brainlest and I please be accurate
joja [24]

Answer:

55.04 m 2

Step-by-step explanation:

I am 100% sure this is right.

please mark me as brainlest

5 0
3 years ago
Read 2 more answers
What fraction of 60 is 18
nordsb [41]
10/3 

hope this helped
3 0
3 years ago
Which ratio can be used to form a proportion with the given ratio? 3 to 4
quester [9]

As soon as I see the word "Which ...", I know that there's a list of choices that
goes along with this question, but you decided not to share them.  So there's
no way for me to help you choose the correct one.

In order to form a proportion with 3/4, you need another fraction (or ratio) that's
equal to 3/4 .  Look for a fraction whose numerator is 3/4 of its denominator. 

Here are a few examples:

-- 6 / 8
-- 24 / 32
-- 300 / 400
-- 12 million / 16 million
-- 3x / 4x
-- 15abc / 20abc
-- 3(x+y+z) / 4(x+y+z)


6 0
2 years ago
What is the difference of 11.790 and 2 1/2
bonufazy [111]

Answer:

9.25 or or 9 1/4

Step-by-step explanation:

7 0
2 years ago
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