The distance between an arbitrary point on the surface and the origin is

Recall that for differentiable functions

and

, the composition

attains extrema at the same points that

does, so we can consider an augmented distance function

The Lagrangian would then be

We have partial derivatives

Set each partial derivative to 0 and solve the system to find the critical points.
From the second equation it follows that either

or

. In the first case we arrive at a contradiction (I'll leave establishing that to you). If

, then we have

This means

so that the points on the surface closest to the origin are

.
90 degrees you are looking to your side
180 degrees you are looking behind you
around origin of 0,0
the image is flipped into the negative world if it is in posiitve or vice versa
Answer:
As 2.551 < 3.84 therefore we reject H0.
As 44.803 > 4.46 so we accept null hypothesis.
Step-by-step explanation:
The answer is attached.
There are three judges so v1 = 3-1= 2 and v2 = (4*2)= 8
There are five gymnasts so v1 = 5-1= 4 and v2= (4*2)= 8
For alpha = 0.05 we find the value of F1 and F2 from the table.
Answer:
8 ounce jar
Step-by-step explanation:
We divide the price by the number of ounces to find the unit rate or cost per one ounce.
2.79/8=0.34875
5.54/15=0.3693
The 8 ounce jar has the best unit rate with a lower cost per ounce.
Let the score of cowboys is x
and giants make score 9 which is twice less than the cowboys score so
giants score will be = 2x -9
and packers scored 14 more than giants that is (2x - 9) + 14
now sum of their scores is equal to 81 it means:
x + (2x - 9) + (2x -9) + 14 = 81
x + 2x - 9 + 2x - 9 + 14 = 81
5x = 81 + 4
5x = 85
x = 17
packers scored = (2x - 9) + 14
= 2 (17) -9 + 14
=38 + 5 = 43 points