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Misc 7 - Find intervals f(x) = x3 + 1/x3 x = 0 is increasing

Misc 7 - Chapter 6 Class 12 Application of Derivatives - Part 2
Misc 7 - Chapter 6 Class 12 Application of Derivatives - Part 3
Misc 7 - Chapter 6 Class 12 Application of Derivatives - Part 4

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Misc 7 Find the intervals in which the function f given by f (x) = x3 + 1/𝑥^3 , 𝑥 ≠ 0 is (i) increasing (ii) decreasing. f(𝑥) = 𝑥3 + 1/𝑥3 Finding f’(𝒙) f’(𝑥) = 𝑑/𝑑𝑥 (𝑥^3+𝑥^(−3) )^. = 3𝑥2 + (−3)^(−3 − 1) = 3𝑥2 – 3𝑥^(−4) = 3𝑥^2−3/𝑥^4 = 3(𝑥^2−1/𝑥^4 ) Putting f’(𝒙) = 0 3(𝑥^2−1/𝑥^4 ) = 0 (𝑥^6 − 1)/𝑥^4 = 0 𝒙^𝟔−𝟏 = 0 (𝑥^3 )^2−(1)^2=0 (𝒙^𝟑−𝟏)(𝒙^𝟑+𝟏)=𝟎 Hence, 𝒙 = 1 & –1 𝑥^3+1 = 0 𝑥3 = −1 𝒙 = −1 Plotting points on number line So, f(𝑥) is strictly increasing on (−∞ , −1) & (1 , ∞) & f(𝑥) strictly decreasing on (−1 , 1) But we need to find Increasing & Decreasing f’(𝑥) = 3(𝑥^2−1/𝑥^4 ) Thus, f(𝑥) is increasing on (−∞ , −𝟏] & [𝟏 , ∞) & f(𝑥) is decreasing on [−𝟏 , 𝟏] For x = −1 f’(−1) = 0 For x = 1 f’(1) = 0

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.