Question

do not press submit until all questions are colored green and you have double checked your work. which of the following statements is/are true regarding the function shown below? i. the function has a vertical asymptote but no horizontal asymptote. ii. the function has a relative maximum at ( x = 0 ). iii. the function has two points of inflection.

Explanation:

To determine which statement is true, we analyze each one:

Statement I: Vertical vs. Horizontal Asymptotes

• A vertical asymptote occurs where the function approaches \( \pm\infty \) (e.g., near \( x = 0 \), the graph shoots up/down, suggesting a vertical asymptote).
• A horizontal asymptote is a horizontal line the function approaches as \( x \to \pm\infty \). From the graph, as \( x \to \pm\infty \), the function does not approach a horizontal line (it continues to curve or extend), so there is no horizontal asymptote. Thus, Statement I is true.

Statement II: Relative Maximum at \( x = 0 \)

A relative maximum is a point where the function changes from increasing to decreasing. At \( x = 0 \), the function has a vertical asymptote (not a smooth point of extrema). The graph does not have a "peak" or "valley" at \( x = 0 \); instead, it has a discontinuity (asymptote). Thus, Statement II is false.

Statement III: Two Points of Inflection

A point of inflection is where the concavity changes (second derivative changes sign). The graph appears to have a single curve (no two distinct points where concavity flips). Thus, Statement III is false.

• Statement I: The graph has a vertical asymptote at \( x = 0 \) (function approaches \( \pm\infty \)) and no horizontal asymptote (no horizontal line as \( x \to \pm\infty \)).
• Statement II: \( x = 0 \) is an asymptote, not a relative maximum (no peak/valley).
• Statement III: The graph does not show two concavity changes (inflection points).

Answer:

I. The function has a vertical asymptote but no horizontal asymptote.