5. Let X, be a RSR (mod n). Prove that E e(E/n) EXn is a multiplicative function of n, where e(x) = e²riz_

Q: x2-a? A B X-a x+a Find A, where a + 0. O a -- a

A: Using partial fraction,

Q: Let {(=) = In ="(z +2)*(z² + 9)] f'(x) =

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Q: (e) If each fn has at most a countable number of discontinuities, then f hasat most a countable…

A: Consider the example: fnx=1q             x= pqwith p,q∈N, gcdp,q=1,q≤n0             otherwise So,…

Q: Exercise 4: How many functions f exist such that f'(x) = 3x2 - 6r +3, and (1,-2) is a point of…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Suppose f"(x) = N;(h) + a,h* + a;h® + azh® + ... When calculating N,(h) using Richardson…

A: Solution

Q: Show that for all x, e* > 1 +x.

A: It is given that, ex≥1+xwe have to show that for all x.

Q: 5.16] Use the function f (x) = xe¯ ¯x² to find fave on [0,4] %3D

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Q: f(x) = -12-21x + 18x^2 - 2.75x^3 what method will give a root of the function equal to - 0.287425 in…

A: We will use a numerical method to determine what will be the value of root of the given function in…

Q: Let F(x) = f(x) = x" for x = [a, b] and some positive odd integer n. Show [₁ 62_a²n 2 fdF =

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Q: 8.5.2 Example B Calculate x given that it satisfies the relation 3" = 5x. (8.193) This latter…

A: We have to explain the given points .

Q: f: ℤ675,683 → ℤ821 × ℤ823 satisfies f(x) = (mod821(x), mod823 (x)). You may use: 1=411⋅821− 410⋅823…

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Q: Define Carmichacl's X-function and compute A(6031).

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Q: The characteristic of Z5[x]/ is O 3 5

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Q: Show that /(-1)-7(1) but there is Let f(x) 1 Show that /(-1) (1) but there is no number e in (-1, 1)…

A:

Q: Prove that x² +¸µ111² +21112 where prime denotes differentiation w.r.t.s. = 1 o² + 1+p¹² P

A:

Q: Which of the following does not satisfy the conditions of the Mean Value Theorem? Select one: O a.…

A: We recall the theorem and then proceed to sovle the problem

Q: How many finite functions exist of the form f: {1, 2, 3,...,m} {1, 2, 3,..., n}? How many of them…

A:

Q: Prove that coth-la = In x +1 for |a| > 1 x - 1

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Q: A container contains 35 green tokens, 5 blue tokens, and 3 red tokens. Two tokens are randomly…

A: There are total (35+5)=40 non-red tokens and 3 red tokens in the container.  And total (35+5+3)=43…

Q: Suppose f: R → R is defined by the property that f(x) = x + x² + x³ for every real number x, and g:…

A: We are given that, for all real number x,          f(x) = x + x2 + x3  And        (g o f)(x) = x…

Q: 7.1 Example: Equality of Functions a. Let J 0,,2}, and define functions fand g from J to J as…

A: (a)Given

Q: 2.59 Find || f |sup if a) f(x) = x on (-1, ). So if x E Q, b) f(x) = – x² .2 if x ¢ Q. -

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Q: Q1/ Let f, = (! ? 3 45 145 3 ). = (2 6 5).f = (1 345 6 2) € p. Find the following: 1) f, of, 2) 5 3)…

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Q: 3. Let h: Z→ Z be the function h(r) = = x mod 4. Find h-¹(0), h-¹({0, 1, 2, 3}), and h-¹(4). Explain…

A: In this question we can easily observe that that output of the function is the remainder when any…

Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

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Q: 2 Let xk = E(k+1)(k+2) for all k e N and define S, = Ek=1 ®k• (a) Evaluate Sn as a function of n.

A:

Q: 2. Let f be the arithmetic function defined by f(n) = (-1)". Compute Z;(2) = f(n) n2 n=1 Hint:…

A: Solution: We will use the following formula Here n0=2

Q: Show that (an) is increasing

A:

Q: Solve for the moment generating function of X if for all natural number n, X ∼ U {1,2,…,n}.

A: Given that, X ∼ U {1,2,…,n} , we need to solve for the moment generating function of X if for all…

Q: Let X~ N(1,4). What is E(X²)?

A: Here, it is given that X~N(1,4) that is X follows Normal distribution with mean μ=1 and variance…

Q: Show that x+1/x is increasing on [1,∞]

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Q: Show that O let @ (x) be the step funetion Show that b x

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Q: The population of a town grows at a rate proportional to the population present at time t. The…

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Q: Show that the following function is decreasing on [1, +∞)

A: Let, f(n)=2-nn Now, to find f'(n) using the quotient rule for derivative…

Q: Prove that logg(h1h2) = logg

A: To prove thatfor all  (where  is the multiplicative group of non-zero elements in the finite field…

Q: 8. Let A CR, y € R, and > 0. Show that m* (SA + y) = 8m* (A).

A:

Q: 4. Let F(r) = In(tan(V37-+12). Write F as a composite of non-composite functions. %3D

A:

Q: Show that f(x) = x* + 2x? + 5x +1 is irreducible over Q.

A: The given polynomial of the function is f(x)=x4+2x2+5x+1 We have to show that the polynomial is…