Question

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

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