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USPshnik [31]
3 years ago
10

Simplify the expression

Mathematics
1 answer:
liq [111]3 years ago
5 0
9a.
(-13.2) + 8.1 = -5.1
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Find the critical points of the function f(x, y) = 8y2x − 8yx2 + 9xy. Determine whether they are local minima, local maxima, or
NARA [144]

Answer:

Saddle point: (0,0)

Local minimum: (\frac{3}{8}, -\frac{3}{8})

Local maxima: (0,-\frac{9}{8}), (\frac{9}{8},0)

Step-by-step explanation:

The function is:

f(x,y) = 8\cdot y^{2}\cdot x -8\cdot y\cdot x^{2} + 9\cdot x \cdot y

The partial derivatives of the function are included below:

\frac{\partial f}{\partial x} = 8\cdot y^{2}-16\cdot y\cdot x+9\cdot y

\frac{\partial f}{\partial x} = y \cdot (8\cdot y -16\cdot x + 9)

\frac{\partial f}{\partial y} = 16\cdot y \cdot x - 8 \cdot x^{2} + 9\cdot x

\frac{\partial f}{\partial y} = x \cdot (16\cdot y - 8\cdot x + 9)

Local minima, local maxima and saddle points are determined by equalizing  both partial derivatives to zero.

y \cdot (8\cdot y -16\cdot x + 9) = 0

x \cdot (16\cdot y - 8\cdot x + 9) = 0

It is quite evident that one point is (0,0). Another point is found by solving the following system of linear equations:

\left \{ {{-16\cdot x + 8\cdot y=-9} \atop {-8\cdot x + 16\cdot y=-9}} \right.

The solution of the system is (3/8, -3/8).

Let assume that y = 0, the nonlinear system is reduced to a sole expression:

x\cdot (-8\cdot x + 9) = 0

Another solution is (9/8,0).

Now, let consider that x = 0, the nonlinear system is now reduced to this:

y\cdot (8\cdot y+9) = 0

Another solution is (0, -9/8).

The next step is to determine whether point is a local maximum, a local minimum or a saddle point. The second derivative test:

H = \frac{\partial^{2} f}{\partial x^{2}} \cdot \frac{\partial^{2} f}{\partial y^{2}} - \frac{\partial^{2} f}{\partial x \partial y}

The second derivatives of the function are:

\frac{\partial^{2} f}{\partial x^{2}} = 0

\frac{\partial^{2} f}{\partial y^{2}} = 0

\frac{\partial^{2} f}{\partial x \partial y} = 16\cdot y -16\cdot x + 9

Then, the expression is simplified to this and each point is tested:

H = -16\cdot y +16\cdot x -9

S1: (0,0)

H = -9 (Saddle Point)

S2: (3/8,-3/8)

H = 3 (Local maximum or minimum)

S3: (9/8, 0)

H = 9 (Local maximum or minimum)

S4: (0, - 9/8)

H = 9 (Local maximum or minimum)

Unfortunately, the second derivative test associated with the function does offer an effective method to distinguish between local maximum and local minimums. A more direct approach is used to make a fair classification:

S2: (3/8,-3/8)

f(\frac{3}{8} ,-\frac{3}{8} ) = - \frac{27}{64} (Local minimum)

S3: (9/8, 0)

f(\frac{9}{8},0) = 0 (Local maximum)

S4: (0, - 9/8)

f(0,-\frac{9}{8} ) = 0 (Local maximum)

Saddle point: (0,0)

Local minimum: (\frac{3}{8}, -\frac{3}{8})

Local maxima: (0,-\frac{9}{8}), (\frac{9}{8},0)

4 0
3 years ago
Valentina has a tray where she keeps all of her important papers. the tray is in the shape of a rectangular prism with no top. a
mel-nik [20]

The volume of a rectangular prism with a length of 12 inches, width of 10 inches, and height of 2.5 inches is 300 cubic inches

<h3>Volume of a rectangular prism</h3>
  • Length= 12 inches
  • Width = 10 inches
  • Height = 2.5 inches

Volume of a rectangular prism = length × width × height

= (12 × 10 × 2.5) inches

= 300 inches

Learn more about volume of prism:

brainly.com/question/23963432

#SPJ4

4 0
2 years ago
ravis gas tank is 1/2 full .after he buys 2 gallons of gas it is 2/3 full how many gallons can ravis tank hold?
aliya0001 [1]
So
1/2x+2=2/3x
convert to common fraction
1/2=3/6
2/3=4/6
3/6x+2=4/6x
subtract 3/6x from both sides
2=1/6x
multiply boh tisdes by 6
12=x
the tank can hold 12 gallon

8 0
3 years ago
Joey sells cars at Orange City Ford. He earns $350 a week plus $200 per car he sells. If he earned $1550 last week, how many car
Vladimir [108]

Answer:

Joey sold 6 cars

Step-by-step explanation:

Joey earns $350 per week + $200 per car he sells

The number of cars that he sold last week is unknown, so represent it with a variable, x.

Write an equation

$350 + $200x = $1,550

Subtract $350 from both sides of this equation

$200x = $1200

Divide both sides of the equation by 200

x = 6

Joey sold 6 cars

Hope this helps :)

7 0
3 years ago
Can someone answer this if you don’t mind
allsm [11]

4x-10 and 7x-8 are supplementary angles and thus add up to 180 degrees.

Thus, 11x - 18 = 180 degrees.

Combining like terms, 11x = 198, and x = 18.

The two supplementary angles are 4(18)-10 and 7(18)-8, or 62 and 118 degrees.

4 0
3 years ago
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