Application of Derivatives

Find the intervals in which the function , is (a) increasing (b) decreasing. - Wired Faculty

Question

Find the intervals in which the function
$straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 straight x squared plus 1 over denominator straight x end fraction comma space space straight x not equal to 0$ , is
(a) increasing (b) decreasing.

Solution

$straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 straight x squared plus 1 over denominator straight x end fraction space equals space 4 straight x plus 1 over straight x therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 minus 1 over straight x squared$
(a) For f(x) to be increasing,
$straight f apostrophe left parenthesis straight x right parenthesis space greater than space 0 space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space 4 minus 1 over straight x squared greater than 0$

rightwards double arrow space space space space space space space space 4 greater than 1 over straight x squared space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space straight x squared greater than 1 fourth space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar squared greater than space open parentheses 1 half close parentheses squared rightwards double arrow space space space space open vertical bar straight x close vertical bar space greater than space 1 half space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space either space straight x less than negative 1 half space space space space space space or space space straight x space greater than space 1 half space therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space space open parentheses negative infinity comma space space space minus 1 half close parentheses space union space open parentheses 1 half comma space space infinity close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

(b) For f(x) to be decreasing,

space space space space space space space space space space space straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space space space space space space space rightwards double arrow space space 4 minus 1 over straight x squared less than 0 space space space space space space space space space space space space space space space space space rightwards double arrow space space 4 straight x squared minus 1 space less than space 0 rightwards double arrow space space space space space space 4 straight x squared less than 1 space space space space space space space space space space rightwards double arrow space space straight x squared less than 1 fourth space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space open vertical bar straight x close vertical bar squared space less than space open parentheses 1 half close parentheses squared rightwards double arrow space space space space space space space space space open vertical bar straight x close vertical bar space less than space 1 half space space space space space space space space space space space space rightwards double arrow space space space space space minus 1 half less than straight x less than 1 half space But space straight x not equal to 0 therefore space space space straight f left parenthesis straight x right parenthesis space is space decreasing space in space open parentheses negative 1 half comma space space 0 close parentheses space space union space open parentheses 0 comma space space 1 half close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

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