First slide

Functions (XII)

Question

The domain of $f (x) = l n (a x^{3} + (a + b) x^{2} + (b + c) x + c),$ where $a > 0, b^{2} - 4 a c = 0,$ is (where [.] represents greatest integer function)

Moderate

A

$(- 1, \infty) - \{- \frac{b}{2 a}\}$
B

$(1, \infty) - \{- \frac{b}{2 a}\}$
C

$(- 1, 1) - \{- \frac{b}{2 a}\}$
D

none of these

Solution

We must have $a x^{3} + (a + b) x^{2} + (b + c) x + c > 0$
$\begin{array}{l} or a x^{2} (x + 1) + b x (x + 1) + c (x + 1) > 0 \\ or (x + 1) (a x^{2} + b x + c) > 0 \\ or a (x + 1) {(x + \frac{b}{2 a})}^{2} > 0 as b^{2} = 4 a c \\ or x > - 1 and x \neq - \frac{b}{2 a} \end{array}$

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