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1. if you start with a statement P, apply some (mathematically sound) operations and arrive at a conclusion Q, which you know to be true, is P definitely true?

is there a general relationship between the roots of f(x)=0 and the roots of f'(x)=0 ?
2. 1. No. For example, you could have P be "0=1", multiply both sides by 0 then get "0=0" as Q.

2. I'm not aware of any nice formula connecting them.
3. what about the opposite? going from a statement to a statement which you know is false - is the original statement false?
4. (Original post by Pheylan)
what about the opposite? going from a statement to a statement which you know is false - is the original statement false?
Yup. Same idea as proof by contradiction. Have a look at some truth tables.
5. (Original post by Clarity Incognito)
Same idea as proof by contradiction
oh yeah
6. (Original post by Pheylan)
is there a general relationship between the roots of f(x)=0 and the roots of f'(x)=0 ?
If we have both f(x) = 0 and f'(x) = 0, then x is called a multiple zero of f. But no, there isn't a general relationship between zeros of f and zeros of f'.
7. A rather obvious thing is that between two roots of f(x) there will be a root of f'(x) viz.

If f(a) = f(b) = 0 with a < b there is a c such that a < c < b and f'(c) = 0.

N.B. this is not stated in an entirely truthful way but for STEP where all functions are sufficiently nice you can use that.

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Updated: December 5, 2010
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