Application of Derivatives

Find the value(s) of x for which y = is an increasing function. - Wired Faculty

Question

Find the value(s) of x for which y = $open square brackets straight x left parenthesis straight x minus 2 right parenthesis close square brackets squared$ is an increasing function.

Solution

Given function is
$straight f left parenthesis straight x right parenthesis space equals space open square brackets straight x left parenthesis straight x minus 2 right parenthesis close square brackets squared rightwards double arrow space straight f apostrophe left parenthesis straight x right parenthesis space equals space straight x squared cross times space 2 left parenthesis straight x minus 2 right parenthesis plus left parenthesis straight x minus 2 right parenthesis squared cross times space 2 straight x rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 straight x left parenthesis straight x minus 2 right parenthesis space open square brackets straight x plus left parenthesis straight x minus 2 right parenthesis close square brackets rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 straight x left parenthesis straight x minus 2 right parenthesis left square bracket 2 straight x minus 2 right square bracket rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 straight x left parenthesis straight x minus 2 right parenthesis thin space open square brackets 2 left parenthesis straight x minus 1 right parenthesis close square brackets rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis space equals 4 straight x left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis$
$Since space straight f apostrophe left parenthesis straight x right parenthesis space is space an space increasing space function comma space straight f apostrophe left parenthesis straight x right parenthesis greater than 0 rightwards double arrow space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 straight x left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x minus 2 right parenthesis greater than 0 rightwards double arrow straight x left parenthesis straight x minus 1 right parenthesis space left parenthesis straight x minus 2 right parenthesis thin space greater than 0 rightwards double arrow 0 less than straight x less than 1 space or space straight x greater than 2 rightwards double arrow straight x space element of space left parenthesis 0 comma space 1 right parenthesis space union space left parenthesis 2 comma space infinity right parenthesis$

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Application Of Derivatives

Find the value(s) of x for which y = $open square brackets straight x left parenthesis straight x minus 2 right parenthesis close square brackets squared$ 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.

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