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3 years ago

8th grade: identify the transformation from the original figure to the image

Mathematics

2 answers:

allsm [11]3 years ago

5 0

I’m being forced to do this please ignore

Delvig [45]3 years ago

5 0

Multiple numbers turned from positive to negative.

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Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Allushta [10]

Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
$f_x=5yz=0\implies y=0\text{ or }z=0$
$f_y=5xz=0\implies x=0\text{ or }z=0$
$f_z=5xy=0\implies x=0\text{ or }y=0$

Taken together, we find that (0, 0, 0) appears to be the only critical point on $f$ within the ball. At this point, we have $f(0,0,0)=0$ .

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

$L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)$

with partial derivatives (set to 0)

$L_x=5yz+2\lambda x=0$
$L_y=5xz+2\lambda y=0$
$L_z=5xy+2\lambda z=0$
$L_\lambda=x^2+y^2+z^2-1=0$

We then observe that

$xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2$

So, ignoring the critical point we've already found at (0, 0, 0),

$5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}$
$5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}$
$5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}$

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of $\left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right)$ , at which points we get a value of either of $\pm\dfrac5{\sqrt3}$ , with the maximum being the positive value and the minimum being the negative one.

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3 years ago

What whole numbers lie between 39

Ludmilka [50]

Answer: none

Step-by-step explanation: there are no WHOLE numbers between 39

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2 years ago

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A student earned a grade of 80% on a math test that had20 problems. How many problems on this test did the student answer correc

Georgia [21]

Answer:

Only 16 problems correct

Step-by-step explanation:

Well I thought that since 80% is 4/5 then I should do 20 divided by 5 which will get me 4 and since I'm doing 4/5 I should go all the way up to 80% of 20 which will get me 16.

I'm not making that much sense but

Hope this helps :)

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2 years ago

What is the hypotenuse of a right triangle?

Alex73 [517]

The hypotenuse^2 = leg1^2 + leg2^2
The hypotenuse is always the LONGEST side of any right triangle.
It is also the side that is opposite the right angle

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3 years ago

What is 5x × 12y - 8xy

makkiz [27]

Answer:

52xy is the answear. Hope it helps

Step-by-step explanation:

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3 years ago

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