7) Find the intervals in which the function f given by f (x) = x ^3 + \frac{1}{x^3}, x \neq 0

i)increasing

Answers (1)

Given function is
f (x) = x ^3 + \frac{1}{x^3}
f^{'} (x) = 3x ^2 + \frac{-3x^2}{x^4}\\ f^{'}(x) = 0\\ 3x ^2 + \frac{-3x^2}{x^4} = 0\\ x^4 = 1\\ x = \pm1

Hence, three intervals are their  (-\infty,-1),(-1,1) \ and (1,\infty)
In interval (-\infty,-1) \ and \ (1,\infty) , f^{'})x > 0
Hence, given function  f (x) = x ^3 + \frac{1}{x^3} is increasing in interval  (-\infty,-1) \ and \ (1,\infty)
In interval (-1,1) , f^{'}(x)< 0
Hence, given  function  f (x) = x ^3 + \frac{1}{x^3}   is decreasing in interval (-1,1)

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