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3 years ago

Is xy = yx? Explain how you know

Mathematics

1 answer:

blagie [28]3 years ago

4 0

Answer:

Yes

Step-by-step explanation:

xy=yx because either way, you would end with the same product. For example, if x was 3 and y was 2, you would have (3)(2)=(2)(3), giving you 6=6 which is, in fact, a true statement.

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Find the extreme values of f subject to both constraints f(x,y) = 2x^2+3y^2-4x-5, x^2 + y^2 <=16

Softa [21]

$f(x,y)=2x^2+3y^2-4x-5$
$f_x=4x-4=0\implies x=1$
$f_y=6y=0\implies y=0$

$f(x,y)$ has only one critical point at $(x,y)=(1,0)$ . The function has Hessian

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}4&0\\0&6\end{bmatrix}$

which is positive definite for all $(x,y)$ , which means $f(x,y)$ attains a minimum at the critical point with a value of $f(1,0)=-7$ .

To find the extrema (if any) along the boundary, parameterize it by $x=4\cos t$ and $y=4\sin t$ , with $0\le t$ . On the boundary, we have

$f(x(t),y(t))=F(t)=2(4\cos t)^2+3(4\sin t)^2-4(4\cos t)-5=32\cos^2t+48\sin^2t-16\cos t-5$
$F(t)=35-16\cos t-8\cos2t$

Find the critical points along the boundary:

$F'(t)=16\sin t+16\sin2t=16\sin t+32\sin t\cos t=16\sin t(1+2\cos t)=0$
$\implies t=0,\dfrac{2\pi}3,\pi,\dfrac{4\pi}3$

Respectively, plugging these values into $F(t)$ gives 11, 47, 43, and 47. We omit the first and third, as we can see the absolute extrema occur when $F(t)=47$ .

Now, solve for $x,y$ for both cases:

$t=\dfrac{2\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=2\sqrt3\end{cases}$

$t=\dfrac{4\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=-2\sqrt3\end{cases}$

so $f(x,y)$ has two absolute maxima at $(x,y)=(-2,\pm2\sqrt3)$ with the same value of 47.

5 0

3 years ago

Write the equation of the line that is parallel to the line y= -2x+3 and goes through the point (3, -6).

svetlana [45]

Answer:

y=-2

Step-by-step explanation:

because it's parallel --> a=-2

therefore we have y=-2x+b

the line goes through the point (3,-6) --> (-2).3+b=-6 --> b=0

3 0

3 years ago

What is 80% of 7.2? i need the answer quickly pleaseee

Natasha2012 [34]

Answer:

i beleive its 5.76

Step-by-step explanation:

4 0

3 years ago

Read 2 more answers

If you spun the spinner 100 times, how many times do you predict that you would spin a number less than 5 on a spinner of 1-6

Roman55 [17]

Answer:

67 times

Step-by-step explanation:

Spinner's sample space = (1, 2, 3, 4, 5, 6)

Number less than 5 = (1, 2, 3, 4)

Total possible outcomes = 6

Required outcome = 4

P(number less than 5) = required outcome / Total possible outcomes

P(number less than 5) = 4 /6 = 2/3

Number of spins = 100

Predicted Number of times a number less than less than 5 is spurn in 100 spins :

100 * 2/3

= 66.666

= 67 times

8 0

3 years ago

56 is 70% of what number

QveST [7]

56=70% of __
56/0.7=80
56 is 70% of 80.

Hope I helped!

5 0

3 years ago