![f(x_1,\ldots,x_n)=x_1+\cdots+x_n=\displaystyle\sum_{i=1}^nx_i](https://tex.z-dn.net/?f=f%28x_1%2C%5Cldots%2Cx_n%29%3Dx_1%2B%5Ccdots%2Bx_n%3D%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5Enx_i)
![{x_1}^2+\cdots+{x_n}^2=\displaystyle\sum_{i=1}^n{x_i}^2=4](https://tex.z-dn.net/?f=%7Bx_1%7D%5E2%2B%5Ccdots%2B%7Bx_n%7D%5E2%3D%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%7Bx_i%7D%5E2%3D4)
The Lagrangian is
![L(x_1,\ldots,x_n,\lambda)=\displaystyle\sum_{i=1}^nx_i+\lambda\left(\sum_{i=1}^n{x_i}^2-4\right)](https://tex.z-dn.net/?f=L%28x_1%2C%5Cldots%2Cx_n%2C%5Clambda%29%3D%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5Enx_i%2B%5Clambda%5Cleft%28%5Csum_%7Bi%3D1%7D%5En%7Bx_i%7D%5E2-4%5Cright%29)
with partial derivatives (all set equal to 0)
![L_{x_i}=1+2\lambda x_i=0\implies x_i=-\dfrac1{2\lambda}](https://tex.z-dn.net/?f=L_%7Bx_i%7D%3D1%2B2%5Clambda%20x_i%3D0%5Cimplies%20x_i%3D-%5Cdfrac1%7B2%5Clambda%7D)
for
, and
![L_\lambda=\displaystyle\sum_{i=1}^n{x_i}^2-4=0](https://tex.z-dn.net/?f=L_%5Clambda%3D%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%7Bx_i%7D%5E2-4%3D0)
Substituting each
into the second sum gives
![\displaystyle\sum_{i=1}^n\left(-\frac1{2\lambda}\right)^2=4\implies\dfrac n{4\lambda^2}=4\implies\lambda=\pm\frac{\sqrt n}4](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%5Cleft%28-%5Cfrac1%7B2%5Clambda%7D%5Cright%29%5E2%3D4%5Cimplies%5Cdfrac%20n%7B4%5Clambda%5E2%7D%3D4%5Cimplies%5Clambda%3D%5Cpm%5Cfrac%7B%5Csqrt%20n%7D4)
Then we get two critical points,
![x_i=-\dfrac1{2\frac{\sqrt n}4}=-\dfrac2{\sqrt n}](https://tex.z-dn.net/?f=x_i%3D-%5Cdfrac1%7B2%5Cfrac%7B%5Csqrt%20n%7D4%7D%3D-%5Cdfrac2%7B%5Csqrt%20n%7D)
or
![x_i=-\dfrac1{2\left(-\frac{\sqrt n}4\right)}=\dfrac2{\sqrt n}](https://tex.z-dn.net/?f=x_i%3D-%5Cdfrac1%7B2%5Cleft%28-%5Cfrac%7B%5Csqrt%20n%7D4%5Cright%29%7D%3D%5Cdfrac2%7B%5Csqrt%20n%7D)
At these points we get a value of
, i.e. a maximum value of
and a minimum value of
.
2-3(z-5)+11=4
<em>Distribute</em>
2-3z+15+11=4
<em>Move the constants to gether but make sure they keep thei signs</em>
2+15+11-3z=4
<em>Simplify</em>
28-3z=4
<em>Start to Isolate the variable by subtracting </em>28<em> from </em><u><em>both</em></u><em> sides of the equation</em>
-3z=-24
<em>Completely isolate the variable by deviding the </em><u><em>whole</em></u><em> equation by </em>-3
z=8
<u><em>If you would like anything explained, just ask</em></u>
Answer:
B. y= -½x + 11
Step-by-step explanation:
_______________
Answer:
150 degrees
Step-by-step explanation:
This image should help. (It was hard to explain, sorry about that ._.)
A full turn is also a
360 degrees turn.
Two thirds is
![\frac { 2 }{ 3 }](https://tex.z-dn.net/?f=%5Cfrac%20%7B%202%20%7D%7B%203%20%7D%20)
To find two thirds of 360 we'll multiply it by
![\frac { 2 }{ 3 }](https://tex.z-dn.net/?f=%5Cfrac%20%7B%202%20%7D%7B%203%20%7D%20)
![360\cdot \frac { 2 }{ 3 } \quad =\quad \frac { 360\cdot 2 }{ 3 } \quad =\quad \frac { 720 }{ 3 } \quad =\quad 240](https://tex.z-dn.net/?f=360%5Ccdot%20%5Cfrac%20%7B%202%20%7D%7B%203%20%7D%20%5Cquad%20%3D%5Cquad%20%5Cfrac%20%7B%20360%5Ccdot%202%20%7D%7B%203%20%7D%20%5Cquad%20%3D%5Cquad%20%5Cfrac%20%7B%20720%20%7D%7B%203%20%7D%20%5Cquad%20%3D%5Cquad%20240)
Two thirds of a full turn (
![{ 360 }^{ o }](https://tex.z-dn.net/?f=%7B%20360%20%7D%5E%7B%20o%20%7D)
) is:
![\boxed { { 240 }^{ o } }](https://tex.z-dn.net/?f=%5Cboxed%20%7B%20%7B%20240%20%7D%5E%7B%20o%20%7D%20%7D%20)
degrees.