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ID:
Class: 12
Subject: Math
Topic: Application of Derivative
Type: Short
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.
Official Solution
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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