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Question

Find the values of x for which y=[x(x2)]2 is an increasing function.

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Solution

y=[x(x2)]2=[x22x]2
dydx=y=2(x22x)(2x2)=4x(x2)(x1)
dydx=0x=0,x=2,x=1
The points x=0,x=1, and x=2 divide the real line into four disjoint intervals
i.e., (,0),(0,1),(1,2) and (2,).
In intervals (,0) and (1,2) , dydx<0
y is strictly decreasing in intervals (,0) and (1,2)
However, in intervals (0,1) and (2,),dydx>0.
y is strictly increasing in intervals (0,1) and (2,).
y is strictly increasing for 0<x<1 and x>2.

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