Create a rational function, g(x) that has the following properties.

Create a rational function, g(x) that has the following properties.

i) V.A.: None (Vertical Asymptote)

ii) O.B.: None (Oblique Asymptote)

iii) H.A (Horizontal Asymptote).: y = 1

iv) Hole: (-4, -3/19)

v) local min.: (-3, - 1/6)

vi) local max.: (1, 1/2)

vii) x-int.: -1

viii) y-int.: 1/3

Solve at a Grade 12 level! Solve using only Grade 12 calculus concepts like derivatives and second derivatives.

A detailed explanation and step by step answer would be highly recommended! Thanks to whoever answers the question!

  • Mathe Mathe

    Bounty seems very low.

    • Savionf Savionf

      I think so too. This is a very time consuming problem and the bounty is very low.

  • @ M F H Could you sumbit the draft? Or whatever work you have so far? Thanks

  • M F H: Please make sure to submit your solution before the deadline.

    • M F H M F H

      Yes of course (else I have to pay again.... :-/) OK, I submit what I have so far but it is getting more complicated than expected at the end... :-(

  • Oh that extra 2.50 was from you not sumbitting the answer in time? No problem I sent it back to you :), sry for the super low bounty I didnt know the problem would be this hard to solve, Im just in grade 12, so I didnt expect my teacher to give such a hard problem, sry lol

    • M F H M F H

      yes it's really long :-( ... I didn't expect it either. No, for the questions we have to pay 80% (so $4 here) to "reserve" them and this is lost if we don't submit an accepted answer :-(

  • M F H M F H

    My approach is based on the requirement that there be no other holes and x-intercepts, as you suggested in the first variant of the question. I show that it is impossible to find such a function with denominator of degree 1+2 or 1+2*2, but for denominator of degree 1+3*2 it should be possible: I find a system of 6 equations for the 6 unknowns which must then be in the denominator of the function, but it is difficult to solve.


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  • M F H M F H

    I'm unable to find a solution, even numerically :-(

  • M F H M F H

    I tried very hard also to find a solution for P(x) = (x+1)² (x²+a1 x + a0) but without success :-(. If your teacher asked this, maybe he gave in his lectures a method to construct such functions ? I will continue to try to find a solution, but it seems that one has to consider an even more general (thus complicated) form of g(x)...

  • ... And the teacher just said in class today, he made a mistake mentioning the horizontal asymptote is when y=0, even though he told otherwise in the email yesterday....... Sorry for making you go through all this problem! :(

  • M F H M F H

    Lol ... OK , yes i think if we don't need to have the h.a. y=1 we have much more liberty in the form of the function.

The answer is accepted.
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