(i) Prove by mathemarical induction on n that 3" > 2n² + 1 for all n € N (ii) Given the function g : R → R defined by if z# -2 g(x) = { ** if 1 = -2 Find whether or not f is injective and surjective. Find the inverse of f, if it exists.

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A: Solution :-

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A: This is a problem of asymptotic growth.

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A: Given:

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A: We can solve this as follows:

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A: Consider the given information:

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A: Step:-1 The Taylor Theorem with Remainder of a function f(x) around a is f(x)=f(a)+…

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A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

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Q: Prove the given theorem and provide (1) examples. Theorem 1.7 (Maclaurin's expansion) It states S(x)…

A: First we write the statement and then proof and one example.

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Q: 05- The distinct residue classes comprising F[x]/f) can now be deseribed explicitly; namely, ups…

A: See answer below.

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Q: Prove that for all x > 0 and all positive integers n x2 e* > 1+x + 2! +3 +: 3! x" + n! Recall that…

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Q: Let f(x) = x In(1+x*). For all æ € [-1, 1], it is known that %3D +o0 In(1+x) = £ (-1)" -xn+1 n+1 n=0…

A: Given: Now, for 7th degree Maclaurin polynomial for f will be,        ln(1+x)=∑n=06(-1)nn+1xn+1. We…

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A: Given that A is a variation of the Ackermann function defined by,A=2nif m=00if m≥1 and n=02if m≥1…

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Q: 3. а) Prove by induction that 1 P(n):2+6+12+20+...+n(2n+2) =n(n+1)(n+ 2) Vn21 1 3 Let f:R→(-1,1) be…

A: For part a, principle of mathematical induction is used. For part b, function is defined when imagee…

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Q: 5 Is the function f(#) = E (-1)" X[n,n+1) n21 integrable on R in the Lebesgue sense. Here XA is the…

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