# Science:Math Exam Resources/Courses/MATH102/December 2012/Question B 02

MATH102 December 2012
Other MATH102 Exams

### Question B 02

Consider the function

${\displaystyle f(x)={\frac {x}{x^{2}+1}}}$

defined on the whole real line.

The zeros of ${\displaystyle \ f'}$, increasing order, are c1 = ___ and c2 = ___.

The zeros of ${\displaystyle \ f''}$, in increasing order, are r1 = ___, r2 = ___ and r3 = ___.

In each empty cell of the tables below, enter a + or - to indicate sign of ${\displaystyle \ f}$, ${\displaystyle \ f'}$ and ${\displaystyle \ f''}$ as appropriate.

 ${\displaystyle (-\infty ,0)}$ 0 ${\displaystyle (0,\infty )}$ ${\displaystyle \ f(x)}$ 0
 ${\displaystyle (-\infty ,c_{1})}$ c1 ${\displaystyle \displaystyle (c_{1},c_{2})}$ c2 ${\displaystyle (c_{2},\infty )}$ ${\displaystyle \ f'(x)}$ 0 0
 ${\displaystyle (-\infty ,r_{1})}$ r1 ${\displaystyle \displaystyle (r_{1},r_{2})}$ r2 ${\displaystyle \displaystyle (r_{2},r_{3})}$ r3 ${\displaystyle (r_{3},\infty )}$ ${\displaystyle \ f''(x)}$ 0 0 0
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