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 = 0

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

The answer is g(x)= (x+4)(x+1)/(x^2+3)(x+4)

But, I'm missing the steps to reach this answer...

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!

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1 Answer

No V.A. means that the numerator is zero when the denominator is.

No O.A. but H.A. y=0 means that lim_{x -> +- oo} f(x) = 0, i.e., numerator must have a lower degree than denominator.

Hole at x=-4 means that f(x) = g(x) (x+4)^k/(x+4)^k where k can be any integer >= 1, and g(x) is well defined in x=-4, and g(x=-4) = the y-value of the hole (-3/19 here).
Everywhere outside x=-4 you can use g(x) instead of f(x), in particular for the extrema (local min / max = points where f'(x) = g'(x) is zero).

So, the simplest choice/try is k=1 and g(x) = (ax+b)/(x² + px + q) where the denominator has no root, i.e., p² - 4q < 0.

x-intercept -1 means that f(x=-1) = 0 <=> g(x=-1) = 0 <=> a = b.

y-intercept 1/3 means that f(x=0) = g(x=0) = 1/3 <=> (0+b)/(0+0+q) = 1/3 <=> q = 3b.

Now to find the remaining unknowns b & p, you must use the equations g(x) = y for each of the given points (x,y). You have 3 points/equations for only 2 unknowns, which is more than enough.

In general you also must compute g'(x) and use g'(x)=0 in the points where you have the local min/max, but here I think you don't need that to find b & p. However, you should maybe check in the end that g'(x)=0 in these two points, since we made the simplifying asumption on the form of g.

(If that wouldn't yield a convenient solution, one would have to try more general functions g(x), i.e., numerator and denominator of higher degree = with additional factors.)

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