Question

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

Solution

$straight y space equals space open square brackets straight x minus left parenthesis straight x minus 2 right parenthesis close square brackets squared space equals space left parenthesis straight x squared minus 2 straight x right parenthesis squared$
$therefore space space space space dy over dx space equals space 2 left parenthesis straight x squared minus 2 straight x right parenthesis. space straight d over dx left parenthesis straight x squared minus 2 straight x right parenthesis space equals space 2 straight x left parenthesis straight x minus 2 right parenthesis thin space left parenthesis 2 straight x minus 2 right parenthesis space equals space 4 straight x left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x minus 2 right parenthesis space space space space space space space space space space dy over dx space equals space 0 space space space give space us space straight x space equals space 0 comma space 1 comma space 2$
The points x = 0, x = 1, x = 2 divide the real line into four disjoining intervals $left parenthesis negative infinity comma space 0 right parenthesis comma space left parenthesis 0 comma space 1 right parenthesis comma space space left parenthesis 1 comma space 2 right parenthesis comma space left parenthesis 2 comma space infinity right parenthesis.$
Now $dy over dx space equals space 4 straight x left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x minus 2 right parenthesis thin space greater than space 0 space in space left parenthesis 0 comma space 1 right parenthesis space and space left parenthesis 2 comma space infinity right parenthesis.$
$therefore space space space space function space straight y space equals space left square bracket straight x left parenthesis straight x minus 2 right parenthesis right square bracket squared$ is increasing in (0, 1) and $left parenthesis 2 comma space infinity right parenthesis.$

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Application Of Derivatives

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

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

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

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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