The rational function has
A) a stationary point at x = 1
B) a stationary point at x = -1
C) two stationary points at approximately x = -1.732 and x = 1.732
D) three stationary points approximately at x = 0, x = -1.732 and x = 1.732
E) no stationary points

Question

The rational function  has
A) a stationary point at x = 1 
B) a stationary point at x = -1 
C) two stationary points at approximately x = -1.732 and x = 1.732 
D) three stationary points approximately at x = 0, x = -1.732 and x = 1.732 
E) no stationary points

Quizplus · Accepted Answer

The given rational function is a square root function with a denominator that causes vertical asymptotes at x = -2 and x = 2. Therefore, the graph has a horizontal branch at y = 0. Since the function is odd, it has symmetry about the origin. The stationary points occur where the derivative is equal to zero. Taking the derivative, we get:

f'(x) = (-6x)/(2(x^2-4)^(3/2))

Setting f'(x) equal to zero gives -6x = 0, which implies x = 0. However, x = 0 is not valid as it's not in the domain of given function.

The other stationary points can be found by setting the denominator equal to zero and solving for x:

x^2 - 4 = 0

(x - 2)(x + 2) = 0

x = -2 or x = 2

Since x = -2 and x = 2 are vertical asymptotes of the function, they do not correspond to stationary points. Hence, the correct choice is C.

The Rational Function Has