Question

1)
For each of the following state whether it is true or false. If
true give one sentence exolanation. If false, give counterexample.

a)If f is differentiable on (0,1) and f is increasing on (0,1)
then f' > 0 on (0,1)

b)Suppose that f is thrice differentiable function defined on
(-1,1). Suppose that the second order Taylor polynomial of f at 0
is 1-x^2. Then f has a local extremum at 0.

Answer #1

NOTE- If it is true, you need to prove it and If it is
false, give a counterexample
f : [a, b] → R is continuous and in the open interval (a,b)
differentiable.
a) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0.
(TRUE or FALSE?)
b) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)
c) If f is reversable, f has no critical point. (TRUE or
FALSE?)
d) If a is a “minimizer”...

Suppose that f and g are infinitely differentiable functions
defined on R. Suppose that Pf is the second order Taylor polynomial
for f centered at 0 and that Pg is the second order Taylor
polynomial for g centered at 0. Let Pfg be the second order Taylor
polynomial for fg centered at 0. Is Pfg = PfPg? If not, is there a
relationship between Pfg and PfPg ?

True or False? Circle whether the following statements are true
or false. Explanations are optional. (a) If f(x) is continuous on
[1, 2], then f(x) attains an absolute maximum on [1, 2]. TRUE FALSE
(b) If f(x) has a local maximum or a local minimum at x = 1, and f
′ (1) exists, then f ′ (1) = 0. TRUE FALSE (c) If f ′ (1) = 0, then
(1, f(1)) is a local maximum or a local minimum...

State, with explanation, whether the following statements are
True or False:
a) It is possible for ? = ?(?) to have an inflection point at
(?, ?(?)) even if ?′(?) is not defined.
b) If ?''(?) > 0 then ? has a local minimum at ? = ?.
c) For all functions ?, if ?′(?) exists ∀?, then ?′′(?) exists
∀?.

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) Let X and Y be sets from Rn. If X ⊂ Y then X is closed if
and only if Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then either X or Y is closed and convex (one or the other).
(iii) Let X be an...

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if
Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then eitherX or Y is closed and convex (one or the other).
(iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.
(iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then
X∪Y...

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...

True or false; for each of the statements below, state whether
they are true or false. If false, give an explanation or example
that illustrates why it's false.
(a) The matrix A = [1 0] is not invertible.
[1 -2]
(b) Let B be a matrix. The rowspaces row (B), row (REF(B)) and
row (RREF(B)) are all equivalent.
(c) Let C be a 5 x 7 matrix with nullity 3. The rank of C is
2.
(d) Let D...

4. For the following claims, state whether they are TRUE or
FALSE. Statements claimed to be TRUE must be accompanied by a
proof, and statements claimed to be FALSE must be accompanied by a
counterexample.
(a) Let A and B be events in the sample space S. Then P(A ∩ B) ≤
P(A)P(B).
(b) Let E be an event with 0 < P(E) < 1 and define PE(A) =
P(A ∩ E) for every event A in the sample space...

1. TRUE or FALSE State whether the statement is true or false,
and also give a brief sentence explaining why you believe this.
(a) If we decrease the confidence level for a fixed n, we
decrease the width of the confidence interval.
(b) A research article reports that a 95% confidence interval
for mean reaction time is from 0.25 to 0.29 seconds. About 95% of
individuals will have reaction times in this interval.
c) In a hypothesis test, a p-value...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 22 minutes ago

asked 25 minutes ago

asked 38 minutes ago

asked 41 minutes ago

asked 52 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago