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) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0.(TRUE or FALSE?)

b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?)
c) If f ′ is limited, then f is lipschitz. (TRUE or FALSE?)

Logic. Identify the form of the following statements:

1. ~ (∃x)((Gx ⊃ Hx) ⋅ (Jx ∨ (Hx ⋅ Kx)))
2. (x)((Gx ⊃ Hx) ⊃ (Jx ∨ Kx))
3. (x)(~ Gx ⊃ Hx) ⊃ (x)(Jx)

2. Consider the plane curve r(t) = <2cos(t),3sin(t)>. Parameterize the osculating circle at t=0. Sketch both the curve and the osculating circle

A. Find all the solutions for the congruence ax ≡ d (mod b) a=123, b=456, d=3

using theorem 11.10 (First Isomorphism Theorem), Show that the set of positive real numbers with multiplication is isomorphic to the set of real numbers with addition.

Theorem 11.10 First Isomorphism Theorem. If ψ : G → H is a group homomorphism with K = kerψ, then K is normal in G. Let ϕ : G → G/K be the canonical homomorphism. Then there exists a unique isomorphism η : G/K → ψ(G) such that ψ = ηϕ.

Find all primes p for which −5 is a quadratic residue.

A=

Diagonalize the matrix above. That is, find matrix D and a nonsingular matrix P such that A = PDP^-1 . Use the representation to find the entries of Aⁿ as a function of n.

Let E/F be a field extension. Let a,b be elements elements of E and algebraic over F. Let m=[Q(a):Q] and n=[Q(b):Q]. Assume that gcd(m,n)=1. Determine the basis of Q(a,b) over Q.

2)

A university finished the season ranked sixth out of 119 teams in football and ninth out of 297 teams in baseball. Based on percentile rank, which team had the better ranking?

A)

the baseball team

Please explain

B)

the football team

Please give a response to below paragraph the one marked response not the question.

The Question:Discuss the properties of some of the commonly used functions on the set of Integers and on the set of Real Numbers. (Ex. Is the exponentiation any or none of Injective, Surjective, Bijective?, etc.)

Response: An exponential function such as exp(x) = e^x is an example of a commonly used injective function, which is a function that is one-to-one, meaning the elements of the domain are not mapped to the same codomain. The example, e^x is not surjective meaning, that the function does not have a right inverse (exp(x)^-1= e^x), in other words, every point in the codomain needs to be mapped to the domain to be considered a surjective function. At first, I was a little confused about what a bijection was as the definition sounds similar to an injective function, but if I understand it right, a bijection refers to a one-to-one correspondence. The difference being that a bijection is a function that is injective and surjective at the same time, and the exponential example (exp(x)=e^x) is not a bijective function since it is injective but not surjective.

Let R^x denote the group of nonzero real numbers under multiplication and let R⁺ denote the group of positive real numbers under multiplication. Let H be the subgroup {1, −1} of R^x. Prove that R^x ≈ R⁺ ⊕ H.

For each given integer n, determine if n is a sum of two squares. (You do not have to find the squares.)

(a) n = 19 (b) n = 45 (c) n = 99 (d) n = 999 (e) n = 1000

a. Show that for all values of ? there is an infinite sequence of positive eigenvalues of the problem

? ′′(?) + ??(?) = 0

? ?(0) + ? ′ (0) = 0, ?(1) = 0 (? = ?????)

b. Find eigenvalues of the problem if ? = 1.