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You can't prove claims about irrational numbers using a calculator! For example 1/(sqrt(10001)-100) is exactly equal to sqrt(10001)+100 THis is easily verified by rationalising the denominator. But when you evalaute on a calculator 1/(sqrt(10001)-100)=200.0050001 and...

I think you meant to say that there is no proof that Pi^4+Pi^5 is not equal to e^6

Actually Pi^4+Pi^5=e^6 is an exact formula Any discrepancies you get when calculating the quantities stem directly from errors in the way the calculator handles Pi and e. This also applies of course to "advanced" packages such as MAPLE MATLAB etc. Calculators and software cannot deal directly...

It's interesting that it could also be a straight line! That would be a brave move in an exam!

What about the rational function f(x)=x/1? or (x-1)/(x-1)?

Exactly! Simple concave down increasing f can have multiple inflection points in e^f!! Makes you appreciate the algebra!

e^f(x) This is a common question and it is important to appreciate that there is no real way to gauge the concavity of the function e^(f(x)) just by considering the graph of f! Consider for example f(x)=2ln(x) g(x)=ln(x) h(x)=(1/2)ln(x) These three graphs are essentially the...