Quadratic lower/upper bound of a function

In summary, the speaker is looking for 2nd order polynomials that can be used to lower or upper bound a given function f(t) in a fixed interval. They have been searching for a solution online but have not been successful. They are specifically interested in functions of the form exp(\alpha t), exp(\alpha t)*cos(\beta t), and exp(\alpha t)*cos^2(\beta t). The speaker is surprised that there is not a systematic approach for finding such bounds.
  • #1
aantam
2
0
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!
 
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  • #2
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
 

FAQ: Quadratic lower/upper bound of a function

What is the quadratic lower bound of a function?

The quadratic lower bound of a function is the minimum value that the function can attain. It is the lowest possible value that the function can output, and it is typically represented by the letter "c" in the quadratic equation y=ax^2+bx+c.

How is the quadratic lower bound of a function calculated?

The quadratic lower bound of a function can be calculated by finding the vertex of the parabola represented by the function. The x-coordinate of the vertex gives the value of the input variable that results in the minimum output value, which is the quadratic lower bound. This can be done by using the formula x=-b/2a, where a and b are the coefficients of the quadratic equation.

Can the quadratic lower bound of a function be negative?

Yes, the quadratic lower bound of a function can be negative. This means that the function has a minimum value that is less than 0. In other words, there are values of the input variable that result in a negative output value, but there is no input value that can produce a lower output value.

What is the purpose of determining the quadratic lower bound of a function?

The purpose of determining the quadratic lower bound of a function is to understand the behavior of the function and its possible range of output values. It can also help in finding the minimum value of the function for optimization problems or in identifying the minimum cost or price of a product or service.

How does the quadratic lower bound of a function compare to its upper bound?

The quadratic lower bound of a function is the minimum value that the function can attain, while the quadratic upper bound is the maximum value that the function can attain. The difference between the two is known as the range of the function. The quadratic lower bound and upper bound are important in understanding the overall behavior and limitations of the function.

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