Find the value(s) of x for which y = [x (x – 2)]^2 is an increasi… - Vidyadip

Question

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

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Answer

$\text{y} = \big[\text{x}(\text{x} - 2)\big]^{2} =\big[\text{x}^{2} - 2 \text{x}\big]^{2}\therefore\frac{\text{dy}}{\text{dx}} = 2 (\text{x}^{2} - 2 \text{x})(2\text{x} - 2 )$
$\Rightarrow\frac{\text{dy}}{\text{dx}} = 4\text{x}(\text{x} - 1)(\text{x} - 2)$
$\frac{\text{dy}}{\text{dx}} = 0 \Rightarrow\text{x} = 0 , \text{x} =1 ,\text{x} = 2 $
$\therefore\text { Intervals are } ( - \infty, 0), (0,1),(1,2)(2,\infty)$
since $\frac{\text{dy}}{\text{dx}} >0\text{ in }(0,1)\text{ or } (2,\infty)$
$\therefore\text{ f(x) is increasing in}(0,1)\bigcup(2, \infty).$

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