Application of Derivatives

Find the intervals in which the function (x + 1) 3 (x – 1) 3 is strictly increasing or decreasing. - Wired Faculty

Question

Find the intervals in which the function (x + 1)³ (x – 1)³ is strictly increasing or decreasing.

Solution

Let $straight f left parenthesis straight x right parenthesis space equals space left parenthesis straight x plus 1 right parenthesis cubed space left parenthesis straight x minus 3 right parenthesis cubed$
$therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis straight x plus 1 right parenthesis cubed. space space straight d over dx open square brackets left parenthesis straight x minus 3 right parenthesis cubed close square brackets plus space left parenthesis straight x minus 3 right parenthesis cubed. space space straight d over dx open square brackets left parenthesis straight x plus 1 right parenthesis cubed close square brackets space space space space space space space space space space space space space space space space space space space space equals left parenthesis straight x plus 1 right parenthesis cubed. space space 3 left parenthesis straight x minus 3 right parenthesis squared space plus space left parenthesis straight x minus 3 right parenthesis to the power of 3. end exponent space 3 space left parenthesis straight x plus 1 right parenthesis squared space space space space space space space space space space space space space space space space space space space equals 3 space left parenthesis straight x plus 1 right parenthesis squared space left parenthesis straight x minus 3 right parenthesis squared space left parenthesis 2 straight x minus 2 right parenthesis space equals space 6 left parenthesis straight x plus 1 right parenthesis squared space left parenthesis straight x minus 3 right parenthesis squared space left parenthesis straight x minus 1 right parenthesis Now comma space space space straight f apostrophe left parenthesis straight x right parenthesis greater than 0 rightwards double arrow space space space space space 6 left parenthesis straight x plus 1 right parenthesis squared space left parenthesis straight x minus 3 right parenthesis squared space left parenthesis straight x minus 1 right parenthesis less than 0 space space space space space space rightwards double arrow space space straight x minus 1 space less than space 0 space space space space space space space rightwards double arrow space space straight x space less than space 1 therefore space space space straight f left parenthesis straight x right parenthesis space is space strictly space decreasing space in space left parenthesis negative infinity comma space 1 right parenthesis. space space space space space space space space space$
$Also comma space straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 rightwards double arrow space space space space space 6 left parenthesis straight x plus 1 right parenthesis squared space left parenthesis straight x minus 3 right parenthesis squared space left parenthesis straight x minus 1 right parenthesis thin space less than 0 space space space space space rightwards double arrow space space space straight x minus 1 space less than space 0 space space space space space rightwards double arrow space space straight x space less than space 1 therefore space space space space straight f left parenthesis straight x right parenthesis space is space strictly space decreasing space in space left parenthesis negative infinity comma space 1 right parenthesis.$

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