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3 years ago

Pls, can you help me whit this? thx♡

Mathematics

1 answer:

aivan3 [116]3 years ago

5 0

Answer:

Step-by-step explanation:

hello : here is an solution

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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

K is an example of.<br> O an equation<br> O a formula<br> O a constant<br> O a variable

Diano4ka-milaya [45]

The answer is (a variable)

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2 years ago

Read 2 more answers

Two angles are supplementary to each other. If the first angle measures 58 then which of the following could be the measure of t

34kurt

Answer: 122°

Step-by-step explanation:

When an angle is supplementary to each other , it means the angle measures to 180°.

Now since one of the two angles is 58°, therefore, the other angle will be

180° - 58°

= 122°

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2 years ago

Locate the mean of 2046,971,3113,and 1850.

Debora [2.8K]

To find the mean, you have to add all the numbers then divide it by how many numbers there are.

In this case, you'll need to add all those numbers and divide it by 4.
$2046+971+3113+1850=7980\\7980\div4=1995$

The mean is 1995.

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3 years ago

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B because pie radius square = area of a circle

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