Polynomial coefficients calculations for inequality in the defined interval

[Eric_2000]

Hi All,

I've a problem and I can't find the answer anywhere. Let we say, that we've some defined interval [ x_a, x_b ] in which values of a function f(x) must be less than y_max (f(x) < y_max) - this are the design criterias. Now the point is, that I want to use any polynomial or rational function to met such assumption. So e.g. practical case: I want to use such a function f(x)=ax⁴+b²x³+cx+d in situation, where in range [0, 10], values of function are less than -20, and beyond this interval - values should be greater. How can I calculate values of a,b,c,d parameters ?

Best Regards,
E.

 

[HallsofIvy]

First, your given polynomial, \(\displaystyle y= ax^4+ b^2x^3+ dx+ e\), looks very peculiar! Did you actually intend that "b" be squared? If b is some arbitrary number, so is \(\displaystyle b^2\) so there isn't really any difference between that and \(\displaystyle y= ax^4+ bx^3+ dx+ e\). And did you really intend not to have no "\(\displaystyle cx^2\)" term? I am inclined to believe that was a typo and you really meant the general fourth degree polynomial, \(\displaystyle y= ax^4+ bx^3+ cx^2+ dx+ e\). Please clarify that.

Now, a polynomial will have its maximum value on a closed interval, like [0, 10], either at an endpoint or where the derivative is 0. In order to have the maximum value of this polynomial, on [0, 10], be less than -20, we must have \(\displaystyle y(0)= e< -20\) and \(\displaystyle y(10)= 10000a+ 1000b+ 100c+ 10d+ e< -20\). Solve the equation \(\displaystyle y'= 4ax^3+ 3bx^2+ 2cx+ d= 0\) to determine where the derivative is 0 and set the value of y less than -20 at each of the solutions. That will be the hard part!

 

[Eric_2000]

Well the form of the proposed polynomial was correct - I mean that - it was a deliberate operation, and there are no typo in it. I just want to know - step by step - how the calcuation procedure looks like, when some interval is defined in which ANY function takes values less or greater than some fixed parameter (like this y_max). Firstly I decided to work with some polynomial function, however I want to work with much more "complicated structures" such as rational functions defining circuits transmitances.

This question born, when I calculated some circuit response (in such a equation some fixed parameters are included with frequency dependent one), and I was wondering what values of this fixed parameters must be to met assumptions related to frequency band.

 

[Jomo]

I just want to know - step by step - how the calculation procedure looks like, when some interval is defined in which ANY function takes values less or greater than some fixed parameter (like this y_max).

I am sorry but we do not offer that type of service here. We prefer that the person making the post do the work with our assistance.

 

[Eric_2000]

Great, however how do I know how to solve something if I don't even know where to look? Could you give me at least the names of the books or links to materials related to such problems ? Mathematics is more my hobby, and I don't have even any mathematical education, so I don't even know what to "hook on".

 

[Jomo]

Can you please explain why you have to look somewhere? I suspect that you mean look online?

 

[Eric_2000]

Sorry, - online.