Determine which of the following is not true for the graph of $f ( x ) = \frac { 4 x } { x ^ { 2 } + 4 }$
A)ƒ is continuous on $( - \infty , \infty )$
B)ƒ has no vertical asymptote.
C)ƒ has a horizontal asymptote: y = 0.
D)ƒ has no point of inflection.
E)ƒ has a relative maximum and a relative minimum.

Question

Determine which of the following is not true for the graph of $f ( x ) = \frac { 4 x } { x ^ { 2 } + 4 }$ 
A)ƒ is continuous on $( - \infty , \infty )$
B)ƒ has no vertical asymptote.
C)ƒ has a horizontal asymptote: y = 0.
D)ƒ has no point of inflection.
E)ƒ has a relative maximum and a relative minimum.

Quizplus · Accepted Answer

The function $f(x) = \frac{4x}{x^2 + 4}$ is continuous everywhere because the denominator $x^2 + 4$ never equals zero for any real number x, ensuring the function does not have any vertical asymptotes and is continuous on $(- \infty, \infty)$. As $x$ approaches infinity, the value of $f(x)$ approaches 0, indicating a horizontal asymptote at $y = 0$. The presence of a relative maximum and minimum can be determined by analyzing the first and second derivatives of $f(x)$, which indicate changes in concavity and the presence of critical points. However, the claim that $f(x)$ has no point of inflection is incorrect because the second derivative test can reveal points where the concavity of the function changes, indicating points of inflection.

Determine Which of the Following Is Not True for the Graph