For the following exercises, consider a third-degree polynomial f(x), which has the properties f ' ( 1 ) = 0 , f ' ( 3 ) = 0 . Determine whether the following statements are true or false. Justify your answer. 246. f ( x ) = 0 for some 1 ≤ x ≤ 3
For the following exercises, consider a third-degree polynomial f(x), which has the properties f ' ( 1 ) = 0 , f ' ( 3 ) = 0 . Determine whether the following statements are true or false. Justify your answer. 246. f ( x ) = 0 for some 1 ≤ x ≤ 3
For the following exercises, consider a third-degree polynomial f(x), which has the properties
f
'
(
1
)
=
0
,
f
'
(
3
)
=
0
. Determine whether the following statements are true or false. Justify your answer.
(^)
k
Recall that for numbers 0 ≤ k ≤ n the binomial coefficient (^) is defined as
n!
k! (n−k)!
Question 1.
(1) Prove the following identity: (22) + (1121) = (n+1).
(2) Use the identity above to prove the binomial theorem by induction. That
is, prove that for any a, b = R,
n
(a + b)" = Σ (^)
an-
n-kyk.
k=0
n
Recall that Σ0 x is short hand notation for the expression x0+x1+
+xn-
(3) Fix x = R, x > 0. Prove Bernoulli's inequality: (1+x)" ≥1+nx, by using
the binomial theorem.
-
Question 2. Prove that ||x| - |y|| ≤ |x − y| for any real numbers x, y.
Question 3. Assume (In) nEN is a sequence which is unbounded above. That is,
the set {xn|nЄN} is unbounded above. Prove that there are natural numbers
N] k for all k Є N.
be natural numbers (nk Є N). Prove that
Question content area top
Part 1
Find the measure of
ABC
for the congruent triangles ABC and
Upper A prime Upper B prime Upper C primeA′B′C′.
79 degrees79°
1533
2930
Part 1
m
ABCequals=enter your response heredegrees
Joy is making Christmas gifts. She has 6 1/12 feet of yarn and will need 4 1/4 to complete our project. How much yarn will she have left over compute this solution in two different ways