Find the intervals in which the function
$ f(x) = x^3 + \frac{1}{x^3}, \quad x \ne 0 $
is (i) Increasing (ii) Decreasing.

Q: Find the intervals in which the function $ f(x) = x^3 + \frac{1}{x^3}, \quad x \ne 0 $ is (i) Increasing (ii) Decreasing.

Explanation: \[ f'(x) = 3x^2 - \frac{3}{x^4} = \frac{3x^6 - 3}{x^4} \] Set derivative to zero: \[ f'(x) = 0 \Rightarrow x^6 = 1 \Rightarrow x = \pm 1 \] Sign of derivative: \[\] $ f'(x) > 0 $ in $ (-\infty, -1) \cup (1, \infty) $ ⟹ Increasing \[\] $ f'(x) < 0 $ in $ (-1, 1) $ ⟹ Decreasing

Question

Find the intervals in which the function 
$ f(x) = x^3 + \frac{1}{x^3}, \quad x \ne 0 $
is (i) Increasing (ii) Decreasing.

Accepted Answer

Explanation: $$
f'(x) = 3x^2 - \frac{3}{x^4} = \frac{3x^6 - 3}{x^4}
$$

Set derivative to zero:
$$
f'(x) = 0 \Rightarrow x^6 = 1 \Rightarrow x = \pm 1
$$

Sign of derivative:
 $ f'(x) > 0 $ in $ (-\infty, -1) \cup (1, \infty) $ ⟹ Increasing
 $ f'(x) < 0 $ in $ (-1, 1) $ ⟹ Decreasing

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