Find the intervals in which the function f given byf(x) = x^ 3 + … - Vidyadip

Question

Find the intervals in which the function f given by$\text{f(x) =} \text{x}^{3} + \frac{1}{\text{x}^{3}}, \text{x} \neq \text{0 is}$ (i) increasing (ii) decreasing.

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Answer

$\text{f'(x)} = 3\text{x}^{2} - \frac{3}{\text{x}^{4}} = \frac{3 (\text{x}^{6} - 1)}{\text{x}^{4}}$$\text{f' (x)} = \text{o} \Rightarrow \text{x} = \pm 1,$
$\therefore \text{possible intervals are} ( -\infty, -1), (-1, 0), (0, 1), (1, \infty)$
$\text{f (x) is increasing in} (- \infty, -1) \text{U} (1, \infty)$
$\text{and decreasing in} (-1,0) \text{U} (0, 1)$

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