Quadratic lower/upper bound of a function

[aantam]

Hi folks,

I have a function f(t), and I want to find 2nd order polynomials that lower/upper bound f(t) in a fixed interval. For instance,

f(t) = exp(2t), 0.1<t<0.4

Find a,b,c so that g(t) = a + b t +c t^2 <f(t) for the given interval

I have been googling for the solution, but apparently no one cares about this problem, although I was expecting it to be already solved :( Anyone could give me a reference to look at? Books, papers, whatever.. Oh, by the way, f(t) does not have to be convex.

Thanks a lot!

 

[aantam]

Sorry, I forgot to mention, I assume some benign conditions for the function so that there always exist a lower and upper bound in the given interval (I could always pick a constant as the bounds but I want something better). Actually, the functions that I have to bound will always be of the form:

exp(\alpha t) , exp(\alpha t)*cos(\beta t), exp(\alpha t)*cos^2(\beta t)

It's weird that I didn't find a systematic approach for finding such bounds..

Thanks again