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____ [38]
3 years ago
8

This extreme value problem has a solution with both a maximum value and a minimum value. use lagrange multipliers to find the ex

treme values of the function subject to the given constraint. f(x, y, z) = 4x + 4y + 2z; 2x2 + 2y2 + 2z2 = 18
Mathematics
1 answer:
Natasha_Volkova [10]3 years ago
4 0

We have ∇f(x,y,z) = ⟨4x3,4y3,4z3⟩ and ∇g(x,y,z) = ⟨2x,2y,2z⟩, so LaGrange’s method gives requires that we solve the following system of equations:

x2 + y2 + z2 <span>= 1
We split into four cases, depending on whether </span>x and y are zero or not:

4x3 = 2λx 4y3 = 2λy 4z3 = 2λz

(1) (2) (3) (4)

(a) x and y are both nonzero. Then equations (1) and (2) tell us that x2 = y2 = λ/2, and putting √√ √√√

this into equations (3) and (4) gives solutions (± 2/2, ± 2/2, 0) and (± 3/3, ± 3/3, ± 3/3). (b) x̸=0buty=0. Thenwehavex2 =λ/2,from(1)andputtingthisinto(4)givesλ/2+z2 =1,

√√ which using (3) gives solutions (±1, 0, 0) and (± 2/2, 0, ± 2/2).

(c) y ̸= 0 but x = 0. This is just like case (b) but with x and y reversed: the solutions are (0, ±1, 0) √√

and (0, ± 2/2, ± 2/<span>2).
(d) </span>x = y = 0. Then equation (4) tells us that z = ±1, so we get the two solutions (0, 0, ±1).

Now we determine which of these points are maxima and minima by simply evaluating f at all these points. We find that the maximum value of f is 1 and occurs at (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1),

√√√√√√

while the minimal value is 1/3, and occurs at (± 3/3, ± 3/3, ± 3/3), (± 3/3, ± 3/3, ± 3/3), √√√

and (± 3/3, ± 3/3, ± 3/<span>3). </span>

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Anestetic [448]

Answer:

The last one

Step-by-step explanation:

FInd a common demominator which is 20 and multiply both so it turns into 4 19/20

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A pizza shop sells three sizes of pizza, and they track how often each size gets ordered along with how much they profit from ea
kari74 [83]

Answer:

μy = $6.56 ; σy = 2.77

Step-by-step explanation:

Given the data :

Mean μx= $8.56

Standard Deviation σx ≈ 2.77

Profit, Y on pizza with current promo :

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μy = μ(x - 2)

μy = μx - $2

μy = $8.56 - $2

μy = $6.56

For the standard deviation of y

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σy = 2.77

8 0
3 years ago
Let f(x)= lxl-3 Write a function g whose graph is a translation 5 units down of the graph of f.
soldier1979 [14.2K]

If the function "f(x) = lxl-3" were translated five units down the graph, g(x) would be lxl-8.

We must be familiar with function transformation and different forms of transformation in order to properly understand the question. When a function is transformed, the graph's curve either "moves to the left/right/up/down," "expands or compresses," or "reflects" to create a new function. For instance, by simply pushing the graph of the function g(x) = x2 up by 7 units, the graph of the function f(x) = x2 + 7 is generated. It is advantageous to convert a function since it saves us from having to create a new function from begin. Function transformations typically fall into one of three categories: 1. 2nd translation 3. dilation Reflection

The given query is about Translation of Function. To create a new function, translation moves the curve up or down and modifies its position. Translation comes in two flavours. Vertical and horizontal translation.

When the curve changes, "the function" shifts upward or downward. By doing this, a function of the form y = f(x) is transformed into f(x) ± k, where k stands for the vertical translation. In this case, the function moves up by k units if k > 0.

The function goes down by 'k' units if k < 0.

The curve in the given problem goes down by 5 units, so k = 5. Which is a vertical translation scenario. Consequently, y = f(x) becomes f(x) - k = g(x) and g(x) = f(x) - k = lxl-3 -5 = lxl - 8.

Therefore the new function after translation is g(x) =  lxl - 8.

Learn more about function and types of transformation such as translation, dilation etc here

brainly.com/question/26092237

#SPJ9

4 0
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Find g (x) - f (x). <br><br> g (x) = x^2 + 1 and f (x) = 2x + 5
allochka39001 [22]

Answer:

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