Unconcrete polynomial value constraints

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Let's say I have this general 3rd order polynomial defined:



$f(x):=ax^3+bx^2+cx+d$



If I have concrete constraints of x and y values, I am able to solve the parameters. For example:



  • $f(0)=0$

  • $f(1)=0$

  • $f'(0)=1$

  • $f'(1)=-1$

gives the solution of $a = 0$, $b = -1$, $c = 1$ and $d = 0$.



Let's say, though, that I do not have such concrete points of the poynomial and its derivatives, and instead a general statement, such as $0 leq f(x) < 5$ for $x in [0, 1] $?



Is it still possible to calculate parameters or a set of parameters that matches these constraints?




polynomials systems-of-equations parametric




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  • 1




    "$f(x) := dots$" is not 3rd order
    – pointguard0
    2 days ago











  • For your example, you could always have $$int_0^1f(x),dxge0$$
    – TheSimpliFire
    2 days ago










  • @pointguard0 fixed
    – Post Self
    2 days ago










  • @TheSimpliFire right, that's a good starting point! I'll see what I can do with it. Thank you very much!
    – Post Self
    2 days ago






  • 1




    @GerryMyerson I am aware of that, but I meant to combine these constraints with actual equations as well so that I get a limited set of these parameters
    – Post Self
    2 days ago














up vote
0
down vote

favorite












Let's say I have this general 3rd order polynomial defined:



$f(x):=ax^3+bx^2+cx+d$



If I have concrete constraints of x and y values, I am able to solve the parameters. For example:



  • $f(0)=0$

  • $f(1)=0$

  • $f'(0)=1$

  • $f'(1)=-1$

gives the solution of $a = 0$, $b = -1$, $c = 1$ and $d = 0$.



Let's say, though, that I do not have such concrete points of the poynomial and its derivatives, and instead a general statement, such as $0 leq f(x) < 5$ for $x in [0, 1] $?



Is it still possible to calculate parameters or a set of parameters that matches these constraints?




polynomials systems-of-equations parametric




share|cite|improve this question

















  • 1




    "$f(x) := dots$" is not 3rd order
    – pointguard0
    2 days ago











  • For your example, you could always have $$int_0^1f(x),dxge0$$
    – TheSimpliFire
    2 days ago










  • @pointguard0 fixed
    – Post Self
    2 days ago










  • @TheSimpliFire right, that's a good starting point! I'll see what I can do with it. Thank you very much!
    – Post Self
    2 days ago






  • 1




    @GerryMyerson I am aware of that, but I meant to combine these constraints with actual equations as well so that I get a limited set of these parameters
    – Post Self
    2 days ago












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let's say I have this general 3rd order polynomial defined:



$f(x):=ax^3+bx^2+cx+d$



If I have concrete constraints of x and y values, I am able to solve the parameters. For example:



  • $f(0)=0$

  • $f(1)=0$

  • $f'(0)=1$

  • $f'(1)=-1$

gives the solution of $a = 0$, $b = -1$, $c = 1$ and $d = 0$.



Let's say, though, that I do not have such concrete points of the poynomial and its derivatives, and instead a general statement, such as $0 leq f(x) < 5$ for $x in [0, 1] $?



Is it still possible to calculate parameters or a set of parameters that matches these constraints?




polynomials systems-of-equations parametric




share|cite|improve this question













Let's say I have this general 3rd order polynomial defined:



$f(x):=ax^3+bx^2+cx+d$



If I have concrete constraints of x and y values, I am able to solve the parameters. For example:



  • $f(0)=0$

  • $f(1)=0$

  • $f'(0)=1$

  • $f'(1)=-1$

gives the solution of $a = 0$, $b = -1$, $c = 1$ and $d = 0$.



Let's say, though, that I do not have such concrete points of the poynomial and its derivatives, and instead a general statement, such as $0 leq f(x) < 5$ for $x in [0, 1] $?



Is it still possible to calculate parameters or a set of parameters that matches these constraints?





polynomials systems-of-equations parametric





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago
























asked 2 days ago









Post Self

1136




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  • 1




    "$f(x) := dots$" is not 3rd order
    – pointguard0
    2 days ago











  • For your example, you could always have $$int_0^1f(x),dxge0$$
    – TheSimpliFire
    2 days ago










  • @pointguard0 fixed
    – Post Self
    2 days ago










  • @TheSimpliFire right, that's a good starting point! I'll see what I can do with it. Thank you very much!
    – Post Self
    2 days ago






  • 1




    @GerryMyerson I am aware of that, but I meant to combine these constraints with actual equations as well so that I get a limited set of these parameters
    – Post Self
    2 days ago












  • 1




    "$f(x) := dots$" is not 3rd order
    – pointguard0
    2 days ago











  • For your example, you could always have $$int_0^1f(x),dxge0$$
    – TheSimpliFire
    2 days ago










  • @pointguard0 fixed
    – Post Self
    2 days ago










  • @TheSimpliFire right, that's a good starting point! I'll see what I can do with it. Thank you very much!
    – Post Self
    2 days ago






  • 1




    @GerryMyerson I am aware of that, but I meant to combine these constraints with actual equations as well so that I get a limited set of these parameters
    – Post Self
    2 days ago







1




1




"$f(x) := dots$" is not 3rd order
– pointguard0
2 days ago





"$f(x) := dots$" is not 3rd order
– pointguard0
2 days ago













For your example, you could always have $$int_0^1f(x),dxge0$$
– TheSimpliFire
2 days ago




For your example, you could always have $$int_0^1f(x),dxge0$$
– TheSimpliFire
2 days ago












@pointguard0 fixed
– Post Self
2 days ago




@pointguard0 fixed
– Post Self
2 days ago












@TheSimpliFire right, that's a good starting point! I'll see what I can do with it. Thank you very much!
– Post Self
2 days ago




@TheSimpliFire right, that's a good starting point! I'll see what I can do with it. Thank you very much!
– Post Self
2 days ago




1




1




@GerryMyerson I am aware of that, but I meant to combine these constraints with actual equations as well so that I get a limited set of these parameters
– Post Self
2 days ago




@GerryMyerson I am aware of that, but I meant to combine these constraints with actual equations as well so that I get a limited set of these parameters
– Post Self
2 days ago















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