Extra Review Problems
Here are the problems I brought to the review session. My suggestion is to
prioritize understanding previous homework problems over attempting these
new problems. I apologize for any typos which may exist. Feel free to email
me about any problem for a hint ([email protected]). Some problems may be
more challenging or more ambiguous than what you would see on an actual
exam. Good luck on Monday!
Inequalities:
1. Give a simple description of
{
(
x,y
)
∈
R
:
x
2
+
xy
+
y
2
≥
0
}
Answer:
R
2
2. For
a >
0, prove the minimum of
ax
2
+
bx
+
c
is attained at
x
=

b
2
a
.
Hint: The inequality you are trying to show is:
ax
2
+
bx
+
c
≤
a
(

b
2
a
)
2
+
b
(

b
2
a
) +
c
. Try getting all terms on one side and factoring.
3. Prove
∀
x,y
∈
R
where
y >
0, we have
x
2
+
y
≥
2
x
√
y
Hint: Make a wise substitution into the AGM inequality
4. Prove
∀
x,y
≥
0,
x
+4
y
4
≥
√
xy
Hint: Make a wise substitution into the AGM inequality
5. A circle is deﬁned by
x
2
+
y
2
+
ax
+
by
=
c
. Show why we need
c
≥ 
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 Spring '07
 GHEORGHICIUC
 Math, Extra Review Problems, AGM inequality, Prove ∀x, previous homework problems

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