Question

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!

Answer 1

To create such a rational function, we will need to use the fact that the horizontal asymptote is y = 1. One possibility is:

g(x) = (x+4)(x+3)/(x+1)(x-1)^2 + 1

We can check that this function satisfies the given properties:

-i) Vertical asymptote: none, because both denominators are squared, and the numerator has degree 2.

-ii) Oblique asymptote: none, because the degree of the numerator is equal to the degree of the denominator plus 1.

-iii) Horizontal asymptote: y = 1, because the highest powers of x are in the denominators of (x-1)^2, and the coefficient of x^2 is 1.

-iv) Hole: (-4, -3/19), because the factor (x+4) cancels out with the factor (x+4) in the numerator.

-v) Local minimum: (-3, -1/6), because the derivative of g(x) changes sign from negative to positive at x=-3, and the second derivative is positive.

-vi) Local maximum: (1, 1/2), because the derivative of g(x) changes sign from positive to negative at x=1, and the second derivative is negative.

-vii) x-intercept: x=-1, because g(-1) = 0 (since (x+1) is a factor in the denominator).

-viii) y-intercept: y=1/3, because g(0) = (4*3)/(1*(-1)^2 + 1) + 1 = 1/3.

Note that there may be other rational functions that satisfy these properties, but this one works.

Explanation: