i. the function has a vertical asymptote but no horizontal asymptote. ii. the function has a relative…

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. a) oi only b) oii and iii only c) oiii only

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. a) oi only b) oii and iii only c) oiii only

Answer

Explanation:

Step1: Analyze asymptotes

As (x) approaches (0), the function goes to positive or negative infinity, so there is a vertical asymptote at (x = 0). As (x\to\pm\infty), the function approaches a non - infinite value (around (y=- 2)), so there is a horizontal asymptote. Thus, statement I is false.

Step2: Analyze relative maximum

At (x = 0), the function has a peak value. The derivative of the function is zero at (x = 0) and the function changes from increasing to decreasing around (x = 0), so the function has a relative maximum at (x = 0). Statement II is true.

Step3: Analyze points of inflection

Points of inflection occur where the second - derivative changes sign. Since the function changes concavity on either side of (x = 0) and at another non - zero (x) value (where the curve changes from concave up to concave down and vice - versa), the function has two points of inflection. Statement III is true.

Answer:

b) II and III only