Application of Derivatives

Find the intervals in which the function f is given by (i) increasing (ii) decreasing - Wired Faculty

Question

Find the intervals in which the function f is given by
$straight f left parenthesis straight x right parenthesis space equals space straight x cubed plus 1 over straight x cubed comma space space space straight x not equal to 0 space space is space$
(i) increasing (ii) decreasing

Solution

Here $straight f left parenthesis straight x right parenthesis space equals space straight x cubed plus 1 over straight x cubed$
$therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 3 straight x squared minus 3 over straight x to the power of 4 space equals space fraction numerator 3 left parenthesis straight x to the power of 6 minus 1 right parenthesis over denominator straight x to the power of 4 end fraction space equals space 3 over straight x to the power of 4 left square bracket left parenthesis straight x squared right parenthesis cubed space minus space left parenthesis 1 right parenthesis cubed right square bracket space space space space space space space space space space space space space space space space space equals 3 over straight x to the power of 4 left square bracket left parenthesis straight x squared minus 1 right parenthesis space left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis right square bracket space equals space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction left parenthesis straight x squared minus 1 right parenthesis$
(i) For $straight f left parenthesis straight x right parenthesis$ to be increasing
$straight f apostrophe left parenthesis straight x right parenthesis space greater than space 0 space space space space space space space rightwards double arrow space space space space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction left parenthesis straight x squared minus 1 right parenthesis space greater than space 0$
$rightwards double arrow space space space straight x squared minus 1 space greater than 0$ $open square brackets because space space space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction greater than 0 close square brackets$
$rightwards double arrow space space space space space straight x squared minus 1 space space space rightwards double arrow space space space open vertical bar straight x close vertical bar squared space greater than space left parenthesis 1 right parenthesis squared space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar space greater than space 1 rightwards double arrow space space space either space straight x space less than space minus 1 space space space or space space space straight x greater than 1$
(ii) For f(x) to be decreasing,
$straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space space space space space rightwards double arrow space space space space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction left parenthesis straight x squared minus 1 right parenthesis thin space less than space 0 rightwards double arrow space space space space straight x squared minus 1 space less than space 0 space space space space rightwards double arrow space space space straight x squared less than 1 space space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar squared space less than space left parenthesis 1 right parenthesis squared space space space rightwards double arrow space space space open vertical bar straight x close vertical bar space less than space 1 rightwards double arrow space space space space space space minus 1 less than straight x less than 1.$

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Application Of Derivatives

Find the intervals in which the function f is given by
$straight f left parenthesis straight x right parenthesis space equals space straight x cubed plus 1 over straight x cubed comma space space space straight x not equal to 0 space space is space$
(i) increasing (ii) decreasing

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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