- Fresh Harbour
- Netherton
- Old Town Harbour
Please mark as brainliest
I'm assuming the constraint involves some plus signs that aren't appearing for some reason, so that you're finding the extrema subject to
![x^2+2y^2+3z^2=96](https://tex.z-dn.net/?f=x%5E2%2B2y%5E2%2B3z%5E2%3D96)
.
Set
![f(x,y,z)=xyz](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Dxyz)
and
![g(x,y,z)=x^2+2y^2+3z^2-96](https://tex.z-dn.net/?f=g%28x%2Cy%2Cz%29%3Dx%5E2%2B2y%5E2%2B3z%5E2-96)
, so that the Lagrangian is
![L(x,y,z,\lambda)=xyz+\lambda(x^2+2y^2+3z^2-96)](https://tex.z-dn.net/?f=L%28x%2Cy%2Cz%2C%5Clambda%29%3Dxyz%2B%5Clambda%28x%5E2%2B2y%5E2%2B3z%5E2-96%29)
Take the partial derivatives and set them equal to zero.
![\begin{cases}L_x=yz+2\lambda x=0\\L_y=xz+4\lambda y=0\\L_z=xy+6\lambda z=0\\L_\lambda=x^2+2y^2+3z^2-96=0\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7DL_x%3Dyz%2B2%5Clambda%20x%3D0%5C%5CL_y%3Dxz%2B4%5Clambda%20y%3D0%5C%5CL_z%3Dxy%2B6%5Clambda%20z%3D0%5C%5CL_%5Clambda%3Dx%5E2%2B2y%5E2%2B3z%5E2-96%3D0%5Cend%7Bcases%7D)
One way to find the possible critical points is to multiply the first three equations by the variable that is missing in the first term and dividing by 2. This gives
![\begin{cases}\dfrac{xyz}2+\lambda x^2=0\\\\\dfrac{xyz}2+2\lambda y^2=0\\\\\dfrac{xyz}2+3\lambda z^2=0\\\\x^2+2y^2+3y^2=96\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%5Cdfrac%7Bxyz%7D2%2B%5Clambda%20x%5E2%3D0%5C%5C%5C%5C%5Cdfrac%7Bxyz%7D2%2B2%5Clambda%20y%5E2%3D0%5C%5C%5C%5C%5Cdfrac%7Bxyz%7D2%2B3%5Clambda%20z%5E2%3D0%5C%5C%5C%5Cx%5E2%2B2y%5E2%2B3y%5E2%3D96%5Cend%7Bcases%7D)
So by adding the first three equations together, you end up with
![\dfrac32xyz+\lambda(x^2+2y^2+3z^2)=0](https://tex.z-dn.net/?f=%5Cdfrac32xyz%2B%5Clambda%28x%5E2%2B2y%5E2%2B3z%5E2%29%3D0)
and the fourth equation allows you to write
![\dfrac32xyz+96\lambda=0\implies \dfrac{xyz}2=-32\lambda](https://tex.z-dn.net/?f=%5Cdfrac32xyz%2B96%5Clambda%3D0%5Cimplies%20%5Cdfrac%7Bxyz%7D2%3D-32%5Clambda)
Now, substituting this into the first three equations in the most recent system yields
![\begin{cases}-32\lambda+\lambda x^2=0\\-32\lambda+2\lambda y^2=0\\-32\lambda+3\lambda z^2=0\end{cases}\implies\begin{cases}x=\pm4\sqrt2\\y=\pm4\\z=\pm4\sqrt{\dfrac23}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D-32%5Clambda%2B%5Clambda%20x%5E2%3D0%5C%5C-32%5Clambda%2B2%5Clambda%20y%5E2%3D0%5C%5C-32%5Clambda%2B3%5Clambda%20z%5E2%3D0%5Cend%7Bcases%7D%5Cimplies%5Cbegin%7Bcases%7Dx%3D%5Cpm4%5Csqrt2%5C%5Cy%3D%5Cpm4%5C%5Cz%3D%5Cpm4%5Csqrt%7B%5Cdfrac23%7D%5Cend%7Bcases%7D)
So we found a grand total of 8 possible critical points. Evaluating
![f(x,y,z)=xyz](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Dxyz)
at each of these points, you find that
![f(x,y,z)](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29)
attains a maximum value of
![\dfrac{128}{\sqrt3}](https://tex.z-dn.net/?f=%5Cdfrac%7B128%7D%7B%5Csqrt3%7D)
whenever exactly none or two of the critical points' coordinates are negative (four cases of this), and a minimum value of
![-\dfrac{128}{\sqrt3}](https://tex.z-dn.net/?f=-%5Cdfrac%7B128%7D%7B%5Csqrt3%7D)
whenever exactly one or all of the critical points' coordinates are negative.
Start with 34,34,34,34,34
minus 5 from 1 number and add 5 to another
29,34,34,34,39
minus 2 from a number and add 2 to another
29,32,36,34,39
minus 10 from a number and add 10 to another
19,32,36,44,39
repeat as many times as needed
basically
if the 5 numbers are x,y,z,t,a then
(x+y+z+t+a)/5=34
times 5 both sides
x+y+z+t+a=170
just find 5 numbers that add to 170
100,20,20,20,5,5 is another one
Answer:
3
Step-by-step explanation:
The actual prime factors of 60 are 2, 3, and 5.
Answer:
The correct option is B. The length of OB' is 3 units.
Step-by-step explanation:
The rule of dilation is D₀,₄. It means dilation by scale factor 4 with center of dilation is origin.
![\text{Scale factor}=\frac{\text{Distance between center and any point of the image}}{\text{Distance between center and corresponding point of the Preimage}}](https://tex.z-dn.net/?f=%5Ctext%7BScale%20factor%7D%3D%5Cfrac%7B%5Ctext%7BDistance%20between%20center%20and%20any%20point%20of%20the%20image%7D%7D%7B%5Ctext%7BDistance%20between%20center%20and%20corresponding%20point%20of%20the%20Preimage%7D%7D)
![\text{Scale factor}=\frac{OB'}{OB}](https://tex.z-dn.net/?f=%5Ctext%7BScale%20factor%7D%3D%5Cfrac%7BOB%27%7D%7BOB%7D)
![4=\frac{OB'}{\frac{3}{4}}](https://tex.z-dn.net/?f=4%3D%5Cfrac%7BOB%27%7D%7B%5Cfrac%7B3%7D%7B4%7D%7D)
![4\times \frac{3}{4}=OB'](https://tex.z-dn.net/?f=4%5Ctimes%20%5Cfrac%7B3%7D%7B4%7D%3DOB%27)
![3=OB'](https://tex.z-dn.net/?f=3%3DOB%27)
The length of OB' is 3 units. Therefore the correct option is option is B.