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Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function. - Mathematics

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Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.

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Solution

f(x)=[x(x2)]2

f'(x)=2[x(x2)]{x2+x}

f'(x)=4x(x2)(x1)

At critical point, f'(x)=0

4x(x2)(x1)=0

x=0,1,2

Interval
f'(x)=4x(x1)(x2)
 
Result
(,0) f'(1)=4(1)(2)(3)=24<0 Decreasing
(0,1) f'(1/2)=4(1/2)(1/2)(3/2)=3/2>0 Increasing
(1,2) f'(3/2)=4(3/2)(1/2)(1/2)=3/2<0 Decreasing
(2,) f'(3)=4(3)(2)(1)=24>0 Increasing

So, the function is increasing in the interval (0,1)(2,).

Concept: Increasing and Decreasing Functions
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