justifying

1. Justifying an extreme value theorem?

I was looking through the coursework document on college board and one sentence says there are three ways to justify an extreme value. What are these three ways?
2. Help in justifying complex number question

Given Z1 and Z2 are two distinct complex numbers. If |Z1| = |Z2|, then Re \frac{z1+ z2}{z1 - z2} = 0 Is the above statement true or false? Show working to justify your answer. i managed to worked it out by letting z1= a+jb and z2= x+jy and then sub it in to work out and got true, not sure if...
3. justifying solutions of an equation

If I wrote ax^2+bx+c=0 --> x= d,e for example, then am I right in thinking that all this says is that d and e are the only possible solutions and that I would need 'iff' to justify that these are the solutions?
4. Justifying the integration of both sides

How can you prove using the definition of anti-differentiation, that if two functions are equal, then their anti-derivatives must differ by a constant? Also can this be used to prove the constant rule for integration??
5. Justifying the closure of a set

My problem is the following: Find \overline{A} \textrm{ for } A=\{(x,\sin(1/x))|x>0.\} I know that \overline{A}=A\cup\{(0,y)|-1\leq y \leq 1,\} but I need to figure out & understand the how'' behind that answer, in other words, give a proof that A-closure is that set above. Thanks.
6. justifying that a definition is "well-defined"

part a of this question was to show that similar matrices have the same trace, which i was able to prove without much difficulty. part b says, how would one define the trace of a linear operator T on a finite dimensional vector space? and how would one show that this definition is well-defined...
7. Justifying the Method of Undetermined Coefficients

I'm trying to "Justify the Method of Undetermined Coefficients" by using the "Annihilator Method." To start, I am given a linear, constant coefficient, nonhomogeneous differential equation: L[y] = g(x) Where L[y] = s_{k}*y^{k} + s_{k-1}*y^{k-1} + ... + s_{0}y [Note: y^{k} means the k'th...