Tips for inequality proofs (1 Viewer)
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yanujw
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Pay attention to what variables are positive, real, what inequalities are initially given etc. as this information is almost certainly neccessary in deducing the inequality
Recognise deducing the inequality as one or more of the following;
1) Adding, subtracting or multiplying given inequalities
2) Using LHS - RHS. This may involve having to put everything into one fraction, then considering the sign on the numerator and denominator.
3) Proving that one side, or LHS - RHS can be factorised (i.e. it is a binomial expansion)
4) Substituting new variables into the variables of a given or already proven inequality
More technical techniques;
4) Reaching a result with two variables, then considering the case where a>b or b>a and proving the inequality holds for either case.
5) Proving that LHS/RHS >1 or <1
6) Multiplying, dividing, adding or subtracting both sides by a certain factor/expression, to reach an expression that is provable by the previous techniques.
7) Performing an operation that does not change the sign of an inequality, such as raising to a power or taking a logarithim of both sides.
There is no need to start randomly trying things immediately - you can take 15 to 30 seconds after looking at a question to think about it before answering, rather than having to scribble out multiple lines.
Recognise deducing the inequality as one or more of the following;
1) Adding, subtracting or multiplying given inequalities
2) Using LHS - RHS. This may involve having to put everything into one fraction, then considering the sign on the numerator and denominator.
3) Proving that one side, or LHS - RHS can be factorised (i.e. it is a binomial expansion)
4) Substituting new variables into the variables of a given or already proven inequality
More technical techniques;
4) Reaching a result with two variables, then considering the case where a>b or b>a and proving the inequality holds for either case.
5) Proving that LHS/RHS >1 or <1
6) Multiplying, dividing, adding or subtracting both sides by a certain factor/expression, to reach an expression that is provable by the previous techniques.
7) Performing an operation that does not change the sign of an inequality, such as raising to a power or taking a logarithim of both sides.
There is no need to start randomly trying things immediately - you can take 15 to 30 seconds after looking at a question to think about it before answering, rather than having to scribble out multiple lines.
netflixwatcher
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I am specifically having trouble with geometric and calculus inequality proofs, mainly starting them. Any tips
In the cambridge excercise which is 2f i am pretty sure that goes over gemomtric and calculus proofs, the questions have many parts so as to guide you on the structure on how to do such question, my advice would be to memorise the sequence of the parts IMO, the first time i saw an AP/GP proof on a circle my brain froze but once you can visualise such proofs they will become a lot easier