Proofs to Grade Problems with this title ask you to analyze

Proofs to Grade. Problems with this title ask you to analyze an alleged proof

of a claim and ask you to give it one of three grades. Assign a grade of A if the

claim and proof are correct, even if the proof is not the simplest or the proof

you would have given. Assign an F if the claim is incorrect, if the main idea

of the proof is incorrect or if most of the statements in it are incorrect. Assign

a C partial credit for a proof that is largely correct, but contains one or two

incorrect statements of justi.cations. Whenever the proof is incorrect, explain

your grade. Tell what is incorrect and why.

1. Claim. If m2 is odd, then m is odd.

\"Proof.\" Assume m is odd. Then m = 2k + 1 for some integer k. Then

m^2 = (2m + 1)^2 = 4m^2 + 4m + 1 = 2(2m^2 + 2m) + 1,

which is odd. Therefore, if m^2 is odd, then m is odd.

2. Claim. For every function f : R --> R, if f has relative minimum or

maximum at x = 0, then f\'0(0) = 0..

\"Proof..The derivative at a relative minimum or maximum must be zero.

3. Claim. If t is an irrational number then 5t is irrational.

\"Proof.\" Assume 5t is rational. Then 5t = p/q where p and q are in-

tegers, q 6= 0. Then t = p/5q which is rational. But t is irrational.

1) The first claim is true

Reason: Let us consider an even number which will be of the form m = 2p

m^2 = 4p^2, which will be even

m = 2p + 1

m^2 = (2p+1)^2 = 4p^2 + 2p + 1 = 2(2p^2+p) + 1. which is an odd number

Hence the claim is correct

2) Claim is False

Consider the function f(x) = |x|, which has relavtive minimum at x=0, but the derivative doesn\'t exist at that point

Hence the claim is false

3) Claim is true

Since the natural number * irrational number will be irrational, hence the claim is correct