09/18/2014, 10:20 PM

Sheldon , when discussing post 9 , you started using second derivatives as well.

what is the motivation , reasoning and justification of that ?

Now I think

if x>0 => f(x) , f ' (x) > 0

then a_n x^n < f(x) - f(0)

is a good equation.

if we additionally have x > 0 => f " (x) > 0

then

a_n x^n < f(x) - f(0)

n a_n x^(n-1) < f ' (x) - f ' (0)

seems a good system of equations.

Analogue if we also have f "' (x) > 0.

etc etc.

We can find extrema of f ' (x) - f ' (0) by considering f " (x).

( the analogue of the classical consideration of f ' (x) to find the min as done in post 9 )

However that does not seem what you had in mind , or was it ?

I feel a bit silly asking this question.

Its probably trivial.

regards

tommy1729

what is the motivation , reasoning and justification of that ?

Now I think

if x>0 => f(x) , f ' (x) > 0

then a_n x^n < f(x) - f(0)

is a good equation.

if we additionally have x > 0 => f " (x) > 0

then

a_n x^n < f(x) - f(0)

n a_n x^(n-1) < f ' (x) - f ' (0)

seems a good system of equations.

Analogue if we also have f "' (x) > 0.

etc etc.

We can find extrema of f ' (x) - f ' (0) by considering f " (x).

( the analogue of the classical consideration of f ' (x) to find the min as done in post 9 )

However that does not seem what you had in mind , or was it ?

I feel a bit silly asking this question.

Its probably trivial.

regards

tommy1729