Find intervals of increasing/decreasing

Chapter 6 Class 12 Application of Derivatives
Concept wise

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Misc 7 Find the intervals in which the function f given by f (x) = x3 + 1/π₯^3 , π₯ β  0 is (i) increasing (ii) decreasing. f(π₯) = π₯3 + 1/π₯3 Finding fβ(π) fβ(π₯) = π/ππ₯ (π₯^3+π₯^(β3) )^. = 3π₯2 + (β3)^(β3 β 1) = 3π₯2 β 3π₯^(β4) = 3π₯^2β3/π₯^4 = 3(π₯^2β1/π₯^4 ) Putting fβ(π) = 0 3(π₯^2β1/π₯^4 ) = 0 (π₯^6 β 1)/π₯^4 = 0 π^πβπ = 0 (π₯^3 )^2β(1)^2=0 (π^πβπ)(π^π+π)=π Hence, π = 1 & β1 π₯^3+1 = 0 π₯3 = β1 π = β1 Plotting points on number line So, f(π₯) is strictly increasing on (ββ , β1) & (1 , β) & f(π₯) strictly decreasing on (β1 , 1) But we need to find Increasing & Decreasing fβ(π₯) = 3(π₯^2β1/π₯^4 ) Thus, f(π₯) is increasing on (ββ , βπ] & [π , β) & f(π₯) is decreasing on [βπ , π] For x = β1 fβ(β1) = 0 For x = 1 fβ(1) = 0