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.
Answer:
1,116
Step-by-step explanation:
2568 + ( - 1452)
= 2568 - 1452 ( because + × - makes -)
= 2568 - 1452
=<u> 1,116</u>
9514 1404 393
Answer:
64k^6 -64k^5 +(80/3)k^4 -(160/27)k^3 +(20/27)k^2 -(4/81)k +1/729
Step-by-step explanation:
The row of Pascal's triangle we need for a 6th power expansion is ...
1, 6, 15, 20, 15, 6, 1
These are the coefficients of the products (a^(n-k))(b^k) in the expansion of (a+b)^n as k ranges from 0 to n.
Your expansion is ...
1(2k)^6(-1/3)^0 +6(2k)^5(-1/3)^1 +15(2k)^4(-1/3)^2 +20(2k)^3(-1/3)^3 +...
15(2k)^2(-1/3)^4 +6(2k)^1(-1/3)^5 +1(2k)^0(-1/3)^6
= 64k^6 -64k^5 +(80/3)k^4 -(160/27)k^3 +(20/27)k^2 -(4/81)k +1/729
Answer:
6 (2 x + 1)
Step-by-step explanation:
Simplify the following:
4 (3 x + 2) - 2
Hint: | Distribute 4 over 3 x + 2.
4 (3 x + 2) = 12 x + 8:
12 x + 8 - 2
Hint: | Group like terms in 8 + 12 x - 2.
Grouping like terms, 8 + 12 x - 2 = 12 x + (8 - 2):
12 x + (8 - 2)
Hint: | Subtract 2 from 8.
8 - 2 = 6:
12 x + 6
Hint: | Factor out the greatest common divisor of the coefficients of 12 x + 6.
Factor 6 out of 12 x + 6:
Answer: 6 (2 x + 1)
Answer:
True.
Step-by-step explanation:
Let a,b,c integers and a | (ab+c), that is to say that there exist a integer k such that ab+c = ka. Then:
ab+c = ka
c = ka-ab
c = a(k-b). So, c is a multiple of a, that is a | c.