Question

Find the values of x for which is an increasing function.

Answer 1

The given function is y= [ x( x−2 ) ] 2 .

Rewrite the above equation.

y= [ x 2 −2x ] 2

Differentiate the function with respect to x.

dy dx = d dx [ x 2 −2x ] 2 =2[ x 2 −2x ]( 2x−2 ) =4x( x−2 )( x−1 )

Substitute dy dx = 0,

4x( x−2 )( x−1 )=0 x=0,2,1

The points x=0, x=1 and x=2 divide the real line into four intervals given by,

( −∞,0 ),( 0,1 ),( 1,2 ),( 2,∞ )

In the interval ( −∞,0 ) and ( 1,2 ),

dy dx <0

In the interval ( 0,1 ) and ( 2,∞ ),

dy dx >0

Thus, the function y= [ x( x−2 ) ] 2 is strictly decreasing in the interval ( −∞,0 ) and ( 1,2 ) and strictly increasing in the interval ( 0,1 ) and ( 2,∞ ).