Question

Let \(f(x)\) be a polynomial of degree three satisfying \(f(0)=-1\) and \(f'(0)=0\). Also, 0 is a stationary point of \(f(x)\). If \(f(x)\) does not have an extremum at \(x=0\), then the value of \[ \int \frac{f(x)}{x^3-1}\, dx \] is

• A. \(\dfrac{x^2}{2}+C\)
• B. \(x+C\)
• C. \(\dfrac{x^3}{6}+C\)
• D. None of these

Hint:

Stationary but not extremum ⇒ point of inflection.

Answer 1

The Correct Option is D

Step 1: Since 0 is stationary but not extremum, \[ f(x)=ax^3-1. \]
Step 2: Hence \[ \int\frac{f(x)}{x^3-1}dx=\int\left(a+\frac{1-a}{x^3-1}\right)dx \] which does not match options (A)–(C).