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grin007 [14]
3 years ago
5

The function f (x comma y )equals 3 xy has an absolute maximum value and absolute minimum value subject to the constraint 3 x sq

uared plus 3 y squared minus 5 xyequals121. Use Lagrange multipliers to find these values.
Mathematics
1 answer:
zmey [24]3 years ago
5 0

Answer:

The maximum value of f is 363, which is reached in (11,11) and (-11,-11) and the minimum value of f is -33, which is reached in (√11,-√11) and (-√11,√11)

Step-by-step explanation:

f(x,y) = 3xy, lets find the gradient of f. First lets compute the derivate of f in terms of x, thinking of y like a constant.

f_x(x,y) = 3y

In a similar way

f_y(x,y) = 3x

Thus,

\nabla{f} = (3y,3x)

The restriction is given by g(x,y) = 121, with g(x,y) = 3x²+3y²-5xy. The partial derivates of g are

[ŧex] g_x(x,y) = 6x-5y [/tex]

g_y(x,y) = 6y - 5x

Thus,

\nabla g(x,y) = (6x-5y,6y-5x)

For the Langrange multipliers theorem, we have that for an extreme (x0,y0) with the restriction g(x,y) = 121, we have that for certain λ,

  • f_x(x_0,y_0) = \lambda \, g_x(x0,y0)
  • f_y(x_0,y_0) = \lambda \, g_y(x_0,y_0)
  • g(x_0,y_0) = 121

This can be translated into

  • 3y = \lambda (6x-5y)
  • 3x = \lambda (-5x+6y)
  • 3 (x_0)^2 + 3(y_0)^2 - 5\,x_0y_0 = 121

If we sum the first two expressions, we obtain

3x + 3y = \lambda (x+y)

Thus, x = -y or λ=3.

If x were -y, then we can replace x for -y in both equations

3y = -11 λ y

-3y = 11 λ y, and therefore

y = 0, or λ = -3/11.

Note that y cant take the value 0 because, since x = -y, we have that x = y = y, and g(x,y) = 0. Therefore, equation 3 wouldnt hold.

Now, lets suppose that λ=3, if that is the case, we can replace in the first 2 equations obtaining

  • 3y = 3(6x-5y) = 18x -15y

thus, 18y = 18x

y = x

and also,

  • 3x = 3(6y-5x) = 18y-15x

18x = 18y

x = y

Therefore, x = y or x = -y.

If x = -y:

Lets evaluate g in (-y,y) and try to find y

g(-y,y) = 3(-y)² + 3y*2 - 5(-y)y = 11y² = 121

Therefore,

y² = 121/11 = 11

y = √11 or y = -√11

The candidates to extremes are, as a result (√11,-√11), (-√11, √11). In both cases, f(x,y) = 3 √11 (-√11) = -33

If x = y:

g(y,y) = 3y²+3y²-5y² = y² = 121, then y = 11 or y = -11

In both cases f(11,11) = f(-11,-11) = 363.

We conclude that the maximum value of f is 363, which is reached in (11,11) and (-11,-11) and the minimum value of f is -33, which is reached in (√11,-√11) and (-√11,√11)

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lubasha [3.4K]

Answer:

  1.  value is 0; x-3 is a factor . . . . . . . . . . . . . .third choice

  2.  evaluates at x = -1; remainder is -11 . . . . first choice

Step-by-step explanation:

Dividing f(x) by (x -a) gives ...

  f(x)/(x -a) = g(x) +r/(x -a) . . . . some quotient and a remainder r

If we multiply this expression by (x -a), we see ...

  f(x) = (x -a)g(x) +r

so

  f(a) = (a -a)g(a) +r . . . . . evaluate the above equation at x=a

  f(a) = 0 +r

  f(a) = r . . . . . . . . . a statement of the remainder theorem

If r=0, then x-a is a factor of f(x) = (x-a)g(x).

___

1. We have "a" = 3, and f(3) = 0. Therefore (x-3) is a factor.

__

2. We have "a" = -1, and f(-1) = -11. Therefore the remainder from division by (x+1) is -11.

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3 years ago
For g(x)=x^2-x find g(x) when x=-2
torisob [31]

Answer:

g(x) = 6

Step-by-step explanation:

Begin with substuting the x variable with -2, we do this because the question has listed the value of x already.

Using the value of x, -2 we determine g(x).

g(x) = -2^2 + 2

Above is what the equation would look as, after you input the value of -2.

Using pemdas, (parantheses, exponents, multiplication, division, addition, subtraction) solve the equation.

-2^2 = 4

Think of it as -2 * -2, which is why -2^2 is 4.

Add 4 +2.

4 + 2 = 6.

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Answer:

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The two person will pay the same amount of money when they both watch 9 hockey games.

<h3>How to find the equation and use it to find cost?</h3>

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A second person pays $14.00 for a ticket to every game but doesn't buy programs. Therefore,

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