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
≥ 
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 GHEORGHICIUC
 Math, Extra Review Problems, AGM inequality, Prove ∀x, previous homework problems

Click to edit the document details