Misc 7 - Chapter 6 Class 12 Application of Derivatives

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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 ) 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 ) 𝑥^3−1 = 0 𝑥^3= –1 𝑥 = 1 𝑥^3+1 = 0 𝑥3 = −1 𝑥 = −1 Putting f’(𝒙) = 0 3(𝑥^2−1/𝑥^4 ) = 0 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 Plotting value of 𝒙 on real line Thus, 𝑥 = –1 & 1 divide the real line into three disjoint intervals i.e. (−∞,−1)(−1, 1)& (1,∞) Hence, f(𝑥) is strictly increasing on (−∞ , −𝟏) & (𝟏 , ∞) & strictly decreasing on (−𝟏 , 𝟏)