i. The Lagrangian is
with critical points whenever
- If
, then 
. - If
, then 
. - Either value of
found above requires that either or , so we get the same critical points as in the previous two cases.
We have 
, 
, 
, and 
, so 
has a minimum value of 9 and a maximum value of 182.25.
ii. The Lagrangian is
with critical points whenever
(because we assume 
)
- If
, then 
. - If
, then 
, and with we have 
.
We have 
, 
, 
, and 
. So has a maximum value of 61 and a minimum value of -60.
(f +G) means to add the two equations together:
5x - 2 + 2x +1
5x + 2x = 7x
-2 + = -1
Answer is C. 7x-1
B is the correct answer. 106
The range is the highest number minus the lowest number. In this case, the highest number is 10 and the lowest number is 6, so the range is 10-6=4
To find the students in the group, simply add up the frequency from each mark. I'll give you an example solution, but I challenge you to do this on your own!
For example, say that there are 4 students with 3 marks and 6 students with 4 marks. There are then 4+6=10 total students in the group
To find the mean, simply add up all up the marks with the amount of frequencies and divide it by the amount of students.
Take our example from earlier - we can write it as a list. Since there are 4 students with 3 marks, we have
3, 3, 3, 3 ( 3 four times)
Since there are 6 students with 4 marks, we have
4, 4, 4, 4, 4, 4 (4 six times)
Add them all up, and divide them by the total amount of students (10) to get
(3+3+3+3+4+4+4+4+4+4)/10=(3*4+4*6)/10=36/10=3.6
Good luck, and feel free to ask questions if you're confused!
