The interval of the decreasing function $f (x) = x^{3} - x^{2} - x - 4$

$(\frac{1}{3}, 1)$
$(- \frac{1}{3}, 1)$
$(- \frac{1}{3}, \frac{1}{3})$
$(- 1, - \frac{1}{3})$

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Solution

The correct option is B
$(- \frac{1}{3}, 1)$

Explanation for correct option:
Find the interval:
Given function is $f (x) = x^{3} - x^{2} - x - 4$
We know that a function $f (x)$ is decreasing when $\frac{d f (x)}{d x} < 0$
Therefore,
$\Rightarrow \frac{d}{d x} (x^{3} - x^{2} - x - 4) < 0 \Rightarrow 3 x^{2} - 2 x - 1 < 0 \Rightarrow 3 x^{2} - 3 x + x - 1 < 0 \Rightarrow 3 x (x - 1) + (x - 1) < 0 \Rightarrow (x - 1) (3 x + 1) < 0$
$\Rightarrow x \in (- \frac{1}{3}, 1)$
Hence, the correct answer is option $(B)$ .

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